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57297781af94a219006aa4a4 | Prime_number | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | The sieve of Eratosthenes would not be valid if what were true? | {
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"if 1 were considered a prime",
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0.35964396595... | The Enlightenment has been frequently linked to the French Revolution of 1789. One view of the political changes that occurred during the Enlightenment is that the "consent of the governed" philosophy as delineated by Locke in Two Treatises of Government (1689) represented a paradigm shift from the old governance paradigm under feudalism known as the "divine right of kings". In this view, the revolutions of the late 1700s and early 1800s were caused by the fact that this governance paradigm shift often could not be resolved peacefully, and therefore violent revolution was the result. Clearly a governance philosophy where the king was never wrong was in direct conflict with one whereby citizens by natural law had to consent to the acts and rulings of their government. | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena. | The Quine-Duhem thesis argues that it's impossible to test a single hypothesis on its own, since each one comes as part of an environment of theories. Thus we can only say that the whole package of relevant theories has been collectively falsified, but cannot conclusively say which element of the package must be replaced. An example of this is given by the discovery of the planet Neptune: when the motion of Uranus was found not to match the predictions of Newton's laws, the theory "There are seven planets in the solar system" was rejected, and not Newton's laws themselves. Popper discussed this critique of naïve falsificationism in Chapters 3 and 4 of The Logic of Scientific Discovery. For Popper, theories are accepted or rejected via a sort of selection process. Theories that say more about the way things appear are to be preferred over those that do not; the more generally applicable a theory is, the greater its value. Thus Newton's laws, with their wide general application, are to be preferred over the much more specific "the solar system has seven planets".[dubious – discuss] | if 1 were considered a prime | 96,611 |
57297781af94a219006aa4a6 | Prime_number | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | What is another function that primes have that the number 1 does not? | {
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"Euler's totient function",
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"the sum of divisors function",
"sum of divisors"
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} | What is another function that primes have that the number [MASK] does not? | [
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0.0588771663... | A large Russian assault on the allied supply base to the southeast, at Balaclava was rebuffed on 25 October 1854.:521–527 The Battle of Balaclava is remembered in the UK for the actions of two British units. At the start of the battle, a large body of Russian cavalry charged the 93rd Highlanders, who were posted north of the village of Kadikoi. Commanding them was Sir Colin Campbell. Rather than 'form square', the traditional method of repelling cavalry, Campbell took the risky decision to have his Highlanders form a single line, two men deep. Campbell had seen the effectiveness of the new Minie rifles, with which his troops were armed, at the Battle of the Alma a month before, and was confident his men could beat back the Russians. His tactics succeeded. From up on the ridge to the west, Times correspondent William Howard Russell saw the Highlanders as a 'thin red streak topped with steel', a phrase which soon became the 'Thin Red Line.' | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena. | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | Euler's totient function | 96,612 |
57297781af94a219006aa4a5 | Prime_number | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | What is one function that prime numbers have that 1 does not? | {
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777,
777,
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773
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"the sum of divisors function",
"relationship of the number to its corresponding value of Euler's totient function",
"relationship of the number to its corresponding value of Euler's totient function",
"relationship of the number to its corresponding value of Euler's totient function",
"the relationship of the number to its corresponding value of Euler's totient function"
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} | What is [MASK] function that prime numbers have that [MASK] does not? | [
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0.0718376711010932... | London is one of the major classical and popular music capitals of the world and is home to major music corporations, such as EMI and Warner Music Group as well as countless bands, musicians and industry professionals. The city is also home to many orchestras and concert halls, such as the Barbican Arts Centre (principal base of the London Symphony Orchestra and the London Symphony Chorus), Cadogan Hall (Royal Philharmonic Orchestra) and the Royal Albert Hall (The Proms). London's two main opera houses are the Royal Opera House and the London Coliseum. The UK's largest pipe organ is at the Royal Albert Hall. Other significant instruments are at the cathedrals and major churches. Several conservatoires are within the city: Royal Academy of Music, Royal College of Music, Guildhall School of Music and Drama and Trinity Laban. | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena. | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | the sum of divisors function | 96,613 |
57297781af94a219006aa4a7 | Prime_number | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | If 1 were to be considered as prime what would the sieve of Eratosthenes yield for all other numbers? | {
"answer_start": [
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682,
659,
579,
579
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"only the single number 1",
"1",
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} | If [MASK] were to be considered as prime what would the sieve of Eratosthenes yield for all other numbers? | [
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0.220222... | METRO began light rail service on January 1, 2004, with the inaugural track ("Red Line") running about 8 miles (13 km) from the University of Houston–Downtown (UHD), which traverses through the Texas Medical Center and terminates at NRG Park. METRO is currently in the design phase of a 10-year expansion plan that will add five more lines. and expand the current Red Line. Amtrak, the national passenger rail system, provides service three times a week to Houston via the Sunset Limited (Los Angeles–New Orleans), which stops at a train station on the north side of the downtown area. The station saw 14,891 boardings and alightings in fiscal year 2008. In 2012, there was a 25 percent increase in ridership to 20,327 passengers embarking from the Houston Amtrak station. | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena. | The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers Z[i], that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z[i], 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not. | only the single number 1 | 96,614 |
572978f91d046914007794d3 | Prime_number | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | What is the name of the Egyptian papyrus that suggests that they may have had knowledge of prime numbers? | {
"answer_start": [
149,
153,
153,
117,
153
],
"text": [
"the Rhind papyrus",
"Rhind",
"Rhind",
"Egyptian fraction",
"Rhind papyrus"
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} | What is the name of the [MASK] papyrus that suggests that they may have had knowledge of prime numbers? | [
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0.356759756803... | The "Notre Dame Victory March" is the fight song for the University of Notre Dame. It was written by two brothers who were Notre Dame graduates. The Rev. Michael J. Shea, a 1904 graduate, wrote the music, and his brother, John F. Shea, who earned degrees in 1906 and 1908, wrote the original lyrics. The lyrics were revised in the 1920s; it first appeared under the copyright of the University of Notre Dame in 1928. The chorus is, "Cheer cheer for old Notre Dame, wake up the echos cheering her name. Send a volley cheer on high, shake down the thunder from the sky! What though the odds be great or small, old Notre Dame will win over all. While her loyal sons are marching, onward to victory!" | Advances in polynomial algebra were made by mathematicians during the Yuan era. The mathematician Zhu Shijie (1249–1314) solved simultaneous equations with up to four unknowns using a rectangular array of coefficients, equivalent to modern matrices. Zhu used a method of elimination to reduce the simultaneous equations to a single equation with only one unknown. His method is described in the Jade Mirror of the Four Unknowns, written in 1303. The opening pages contain a diagram of Pascal's triangle. The summation of a finite arithmetic series is also covered in the book. | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | the Rhind papyrus | 96,615 |
572978f91d046914007794d4 | Prime_number | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | What civilization was the first known to clearly study prime numbers? | {
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"the Ancient Greeks",
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} | What civilization was the [MASK] known to clearly study prime numbers? | [
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-0.0176674034... | The Dean of the College of Cardinals, or Cardinal-dean, is the primus inter pares of the College of Cardinals, elected by the cardinal bishops holding suburbicarian sees from among their own number, an election, however, that must be approved by the Pope. Formerly the position of dean belonged by right to the longest-serving of the cardinal bishops. | Advances in polynomial algebra were made by mathematicians during the Yuan era. The mathematician Zhu Shijie (1249–1314) solved simultaneous equations with up to four unknowns using a rectangular array of coefficients, equivalent to modern matrices. Zhu used a method of elimination to reduce the simultaneous equations to a single equation with only one unknown. His method is described in the Jade Mirror of the Four Unknowns, written in 1303. The opening pages contain a diagram of Pascal's triangle. The summation of a finite arithmetic series is also covered in the book. | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | the Ancient Greeks | 96,616 |
572978f91d046914007794d5 | Prime_number | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | What work from around 300 BC has significant theorems about prime numbers? | {
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349
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"text": [
"Euclid's Elements",
"Euclid's Elements",
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"Euclid's Elements",
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} | What work from [MASK] has significant theorems about prime numbers? | [
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-0.01398124... | There are two methods of corruption of the judiciary: the state (through budget planning and various privileges), and the private. Budget of the judiciary in many transitional and developing countries is almost completely controlled by the executive. The latter undermines the separation of powers, as it creates a critical financial dependence of the judiciary. The proper national wealth distribution including the government spending on the judiciary is subject of the constitutional economics. Judicial corruption can be difficult to completely eradicate, even in developed countries. | Advances in polynomial algebra were made by mathematicians during the Yuan era. The mathematician Zhu Shijie (1249–1314) solved simultaneous equations with up to four unknowns using a rectangular array of coefficients, equivalent to modern matrices. Zhu used a method of elimination to reduce the simultaneous equations to a single equation with only one unknown. His method is described in the Jade Mirror of the Four Unknowns, written in 1303. The opening pages contain a diagram of Pascal's triangle. The summation of a finite arithmetic series is also covered in the book. | Virgil's biographical tradition is thought to depend on a lost biography by Varius, Virgil's editor, which was incorporated into the biography by Suetonius and the commentaries of Servius and Donatus, the two great commentators on Virgil's poetry. Although the commentaries no doubt record much factual information about Virgil, some of their evidence can be shown to rely on inferences made from his poetry and allegorizing; thus, Virgil's biographical tradition remains problematic. | Euclid's Elements | 96,617 |
572978f91d046914007794d6 | Prime_number | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | Who demonstrated how to create a perfect number from a Mersenne prime? | {
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501
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"text": [
"Euclid",
"Euclid",
"Euclid",
"Euclid",
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} | Who demonstrated how to create a perfect number from a [MASK] prime? | [
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0.1015381589... | In the city, the population was spread out with 21.9% at age 19 and under, 14.3% from 20 to 24, 33.2% from 25 to 44, 20.4% from 45 to 64, and 10.1% who were 65 years of age or older. The median age was 30.8 years. For every 100 females, there were 92.0 males. For every 100 females age 18 and over, there were 89.9 males. There were 252,699 households, of which 20.4% had children under the age of 18 living in them, 25.5% were married couples living together, 16.3% had a female householder with no husband present, and 54.0% were non-families. 37.1% of all households were made up of individuals and 9.0% had someone living alone who was 65 years of age or older. The average household size was 2.26 and the average family size was 3.08. | Advances in polynomial algebra were made by mathematicians during the Yuan era. The mathematician Zhu Shijie (1249–1314) solved simultaneous equations with up to four unknowns using a rectangular array of coefficients, equivalent to modern matrices. Zhu used a method of elimination to reduce the simultaneous equations to a single equation with only one unknown. His method is described in the Jade Mirror of the Four Unknowns, written in 1303. The opening pages contain a diagram of Pascal's triangle. The summation of a finite arithmetic series is also covered in the book. | Ancient tables provided the sun's mean longitude. Christopher Clavius, the architect of the Gregorian calendar, noted that the tables agreed neither on the time when the sun passed through the vernal equinox nor on the length of the mean tropical year. Tycho Brahe also noticed discrepancies. The Gregorian leap year rule (97 leap years in 400 years) was put forward by Petrus Pitatus of Verona in 1560. He noted that it is consistent with the tropical year of the Alfonsine tables and with the mean tropical year of Copernicus (De revolutionibus) and Reinhold (Prutenic tables). The three mean tropical years in Babylonian sexagesimals as the excess over 365 days (the way they would have been extracted from the tables of mean longitude) were 14,33,9,57 (Alphonsine), 14,33,11,12 (Copernicus) and 14,33,9,24 (Reinhold). All values are the same to two places (14:33) and this is also the mean length of the Gregorian year. Thus Pitatus' solution would have commended itself to the astronomers. | Euclid | 96,618 |
572978f91d046914007794d7 | Prime_number | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | What does the Sieve of Eratosthenes do? | {
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} | What does the Sieve of Eratosthenes do? | [
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0.0724959000945... | Communications in Somalia encompasses the communications services and capacity of Somalia. Telecommunications, internet, radio, print, television and postal services in the nation are largely concentrated in the private sector. Several of the telecom firms have begun expanding their activities abroad. The Federal government operates two official radio and television networks, which exist alongside a number of private and foreign stations. Print media in the country is also progressively giving way to news radio stations and online portals, as internet connectivity and access increases. Additionally, the national postal service is slated to be officially relaunched in 2013 after a long absence. In 2012, a National Communications Act was also approved by Cabinet members, which lays the foundation for the establishment of a National Communications regulator in the broadcasting and telecommunications sectors. | Advances in polynomial algebra were made by mathematicians during the Yuan era. The mathematician Zhu Shijie (1249–1314) solved simultaneous equations with up to four unknowns using a rectangular array of coefficients, equivalent to modern matrices. Zhu used a method of elimination to reduce the simultaneous equations to a single equation with only one unknown. His method is described in the Jade Mirror of the Four Unknowns, written in 1303. The opening pages contain a diagram of Pascal's triangle. The summation of a finite arithmetic series is also covered in the book. | The Antikythera mechanism is believed to be the earliest mechanical analog "computer", according to Derek J. de Solla Price. It was designed to calculate astronomical positions. It was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, and has been dated to circa 100 BC. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later. | compute primes | 96,619 |
57297a276aef051400154f88 | Prime_number | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | In what year did Pierre de Fermat declare Fermat's little theorem? | {
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0.4683212041... | Controversy erupted when Madonna decided to adopt from Malawi again. Chifundo "Mercy" James was finally adopted in June 2009. Madonna had known Mercy from the time she went to adopt David. Mercy's grandmother had initially protested the adoption, but later gave in, saying "At first I didn't want her to go but as a family we had to sit down and reach an agreement and we agreed that Mercy should go. The men insisted that Mercy be adopted and I won't resist anymore. I still love Mercy. She is my dearest." Mercy's father was still adamant saying that he could not support the adoption since he was alive. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | Although Max insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics. | In 1640 | 96,620 |
57297a276aef051400154f89 | Prime_number | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | Besides Leibniz, what other mathematician proved the validity of Fermat's little theorem? | {
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0.06804779171... | The median income for a household in the city was $26,969, and the median income for a family was $31,997. Males had a median income of $25,471 versus $23,863 for females. The per capita income for the city was $15,402. About 19.1% of families and 23.6% of the population were below the poverty line, including 29.1% of those under age 18 and 18.9% of those age 65 or over. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and geometry, thanks to David Hilbert. At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class. | Euler | 96,621 |
57297a276aef051400154f8a | Prime_number | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | Of what form do Fermat numbers take? | {
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0.34608376026153... | Everton F.C. is a limited company with the board of directors holding a majority of the shares. The club's most recent accounts, from May 2014, show a net total debt of £28.1 million, with a turnover of £120.5 million and a profit of £28.2 million. The club's overdraft with Barclays Bank is secured against the Premier League's "Basic Award Fund", a guaranteed sum given to clubs for competing in the Premier League. Everton agreed a long-term loan of £30 million with Bear Stearns and Prudential plc in 2002 over the duration of 25 years; a consolidation of debts at the time as well as a source of capital for new player acquisitions. Goodison Park is secured as collateral. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | 22n + 1 | 96,622 |
57297a276aef051400154f8c | Prime_number | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | Of what form do Mersenne primes take? | {
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0.188582018... | In international law and international relations, a protocol is generally a treaty or international agreement that supplements a previous treaty or international agreement. A protocol can amend the previous treaty, or add additional provisions. Parties to the earlier agreement are not required to adopt the protocol. Sometimes this is made clearer by calling it an "optional protocol", especially where many parties to the first agreement do not support the protocol. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | 2p − 1 | 96,623 |
57297a276aef051400154f8b | Prime_number | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | To what extent did Fermat confirm the validity of Fermat numbers? | {
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0.1176761984825... | Initially the existing 5:3 aspect ratio had been the main candidate but, due to the influence of widescreen cinema, the aspect ratio 16:9 (1.78) eventually emerged as being a reasonable compromise between 5:3 (1.67) and the common 1.85 widescreen cinema format. An aspect ratio of 16:9 was duly agreed upon at the first meeting of the IWP11/6 working party at the BBC's Research and Development establishment in Kingswood Warren. The resulting ITU-R Recommendation ITU-R BT.709-2 ("Rec. 709") includes the 16:9 aspect ratio, a specified colorimetry, and the scan modes 1080i (1,080 actively interlaced lines of resolution) and 1080p (1,080 progressively scanned lines). The British Freeview HD trials used MBAFF, which contains both progressive and interlaced content in the same encoding. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time. | up to n = 4 (or 216 + 1) | 96,624 |
57297bc9af94a219006aa4c7 | Prime_number | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | What is the most elemental way to test the primality of any integer n? | {
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0.0702835991978... | The Panthers beat the Seattle Seahawks in the divisional round, running up a 31–0 halftime lead and then holding off a furious second half comeback attempt to win 31–24, avenging their elimination from a year earlier. The Panthers then blew out the Arizona Cardinals in the NFC Championship Game, 49–15, racking up 487 yards and forcing seven turnovers. | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | trial division | 96,625 |
57297bc9af94a219006aa4c8 | Prime_number | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | What makes the method of trial division more efficient? | {
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0.2209105342... | Monopole antennas consist of a single radiating element such as a metal rod, often mounted over a conducting surface, a ground plane. One side of the feedline from the receiver or transmitter is connected to the rod, and the other side to the ground plane, which may be the Earth. The most common form is the quarter-wave monopole which is one-quarter of a wavelength long and has a gain of 5.12 dBi when mounted over a ground plane. Monopoles have an omnidirectional radiation pattern, so they are used for broad coverage of an area, and have vertical polarization. The ground waves used for broadcasting at low frequencies must be vertically polarized, so large vertical monopole antennas are used for broadcasting in the MF, LF, and VLF bands. Small monopoles are used as nondirectional antennas on portable radios in the HF, VHF, and UHF bands. | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann. | if a complete list of primes up to is known | 96,626 |
57297bc9af94a219006aa4c9 | Prime_number | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | Trial division involves dividing n by every integer m greater than what? | {
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} | Trial division involves dividing n by every integer m greater than what? | [
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0.3082560002803802... | There are many notable contributors to the field of Chinese science throughout the ages. One of the best examples would be Shen Kuo (1031–1095), a polymath scientist and statesman who was the first to describe the magnetic-needle compass used for navigation, discovered the concept of true north, improved the design of the astronomical gnomon, armillary sphere, sight tube, and clepsydra, and described the use of drydocks to repair boats. After observing the natural process of the inundation of silt and the find of marine fossils in the Taihang Mountains (hundreds of miles from the Pacific Ocean), Shen Kuo devised a theory of land formation, or geomorphology. He also adopted a theory of gradual climate change in regions over time, after observing petrified bamboo found underground at Yan'an, Shaanxi province. If not for Shen Kuo's writing, the architectural works of Yu Hao would be little known, along with the inventor of movable type printing, Bi Sheng (990-1051). Shen's contemporary Su Song (1020–1101) was also a brilliant polymath, an astronomer who created a celestial atlas of star maps, wrote a pharmaceutical treatise with related subjects of botany, zoology, mineralogy, and metallurgy, and had erected a large astronomical clocktower in Kaifeng city in 1088. To operate the crowning armillary sphere, his clocktower featured an escapement mechanism and the world's oldest known use of an endless power-transmitting chain drive. | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | greater than 1 | 96,627 |
57297bc9af94a219006aa4cb | Prime_number | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | How many divisions are required to verify the primality of the number 37? | {
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0.2321806997060775... | Napoleon's use of propaganda contributed to his rise to power, legitimated his régime, and established his image for posterity. Strict censorship, controlling aspects of the press, books, theater, and art, was part of his propaganda scheme, aimed at portraying him as bringing desperately wanted peace and stability to France. The propagandistic rhetoric changed in relation to events and to the atmosphere of Napoleon's reign, focusing first on his role as a general in the army and identification as a soldier, and moving to his role as emperor and a civil leader. Specifically targeting his civilian audience, Napoleon fostered an important, though uneasy, relationship with the contemporary art community, taking an active role in commissioning and controlling different forms of art production to suit his propaganda goals. | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time. | only three divisions | 96,628 |
57297bc9af94a219006aa4ca | Prime_number | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | What must the integer m be less than or equal to when performing trial division? | {
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0.1778222918... | Georgian succeeded the English Baroque of Sir Christopher Wren, Sir John Vanbrugh, Thomas Archer, William Talman, and Nicholas Hawksmoor; this in fact continued into at least the 1720s, overlapping with a more restrained Georgian style. The architect James Gibbs was a transitional figure, his earlier buildings are Baroque, reflecting the time he spent in Rome in the early 18th century, but he adjusted his style after 1720. Major architects to promote the change in direction from baroque were Colen Campbell, author of the influential book Vitruvius Britannicus (1715-1725); Richard Boyle, 3rd Earl of Burlington and his protégé William Kent; Isaac Ware; Henry Flitcroft and the Venetian Giacomo Leoni, who spent most of his career in England. Other prominent architects of the early Georgian period include James Paine, Robert Taylor, and John Wood, the Elder. The European Grand Tour became very common for wealthy patrons in the period, and Italian influence remained dominant, though at the start of the period Hanover Square, Westminster (1713 on), developed and occupied by Whig supporters of the new dynasty, seems to have deliberately adopted German stylisic elements in their honour, especially vertical bands connecting the windows. | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | If the input size is n, the time taken can be expressed as a function of n. Since the time taken on different inputs of the same size can be different, the worst-case time complexity T(n) is defined to be the maximum time taken over all inputs of size n. If T(n) is a polynomial in n, then the algorithm is said to be a polynomial time algorithm. Cobham's thesis says that a problem can be solved with a feasible amount of resources if it admits a polynomial time algorithm. | less than or equal to the square root of n | 96,629 |
57297d421d046914007794e5 | Prime_number | Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite. | How many modern types of primality tests for general numbers n are there? | {
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0.1803254038095474... | In the race for individual contributions, economist Lyndon LaRouche dominated the pack leading up to the primaries. According to the Federal Election Commission statistics, LaRouche had more individual contributors to his 2004 presidential campaign than any other candidate, until the final quarter of the primary season, when John Kerry surpassed him. As of the April 15 filing, LaRouche had 7834 individual contributions, of those who have given cumulatively, $200 or more, as compared to 6257 for John Kerry, 5582 for John Edwards, 4090 for Howard Dean, and 2744 for Gephardt. | A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm. | The complexity class P is often seen as a mathematical abstraction modeling those computational tasks that admit an efficient algorithm. This hypothesis is called the Cobham–Edmonds thesis. The complexity class NP, on the other hand, contains many problems that people would like to solve efficiently, but for which no efficient algorithm is known, such as the Boolean satisfiability problem, the Hamiltonian path problem and the vertex cover problem. Since deterministic Turing machines are special non-deterministic Turing machines, it is easily observed that each problem in P is also member of the class NP. | two main classes | 96,630 |
57297d421d046914007794e6 | Prime_number | Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite. | What is the name of one type of modern primality test? | {
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0.021497696... | Xiamen dialect, sometimes known as Amoy, is the main dialect spoken in the Chinese city of Xiamen and its surrounding regions of Tong'an and Xiang'an, both of which are now included in the Greater Xiamen area. This dialect developed in the late Ming dynasty when Xiamen was increasingly taking over Quanzhou's position as the main port of trade in southeastern China. Quanzhou traders began travelling southwards to Xiamen to carry on their businesses while Zhangzhou peasants began traveling northwards to Xiamen in search of job opportunities. It is at this time when a need for a common language arose. The Quanzhou and Zhangzhou varieties are similar in many ways (as can be seen from the common place of Henan Luoyang where they originated), but due to differences in accents, communication can be a problem. Quanzhou businessmen considered their speech to be the prestige accent and considered Zhangzhou's to be a village dialect. Over the centuries, dialect leveling occurred and the two speeches mixed to produce the Amoy dialect. | A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm. | The question of whether P equals NP is one of the most important open questions in theoretical computer science because of the wide implications of a solution. If the answer is yes, many important problems can be shown to have more efficient solutions. These include various types of integer programming problems in operations research, many problems in logistics, protein structure prediction in biology, and the ability to find formal proofs of pure mathematics theorems. The P versus NP problem is one of the Millennium Prize Problems proposed by the Clay Mathematics Institute. There is a US$1,000,000 prize for resolving the problem. | probabilistic (or "Monte Carlo") | 96,631 |
57297d421d046914007794e7 | Prime_number | Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite. | What is the name of another type of modern primality test? | {
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0.0543056316... | The British high-definition TV service started trials in August 1936 and a regular service on 2 November 1936 using both the (mechanical) Baird 240 line sequential scan (later to be inaccurately rechristened 'progressive') and the (electronic) Marconi-EMI 405 line interlaced systems. The Baird system was discontinued in February 1937. In 1938 France followed with their own 441-line system, variants of which were also used by a number of other countries. The US NTSC 525-line system joined in 1941. In 1949 France introduced an even higher-resolution standard at 819 lines, a system that should have been high definition even by today's standards, but was monochrome only and the technical limitations of the time prevented it from achieving the definition of which it should have been capable. All of these systems used interlacing and a 4:3 aspect ratio except the 240-line system which was progressive (actually described at the time by the technically correct term "sequential") and the 405-line system which started as 5:4 and later changed to 4:3. The 405-line system adopted the (at that time) revolutionary idea of interlaced scanning to overcome the flicker problem of the 240-line with its 25 Hz frame rate. The 240-line system could have doubled its frame rate but this would have meant that the transmitted signal would have doubled in bandwidth, an unacceptable option as the video baseband bandwidth was required to be not more than 3 MHz. | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes. | deterministic | 96,632 |
57297d421d046914007794e8 | Prime_number | Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite. | What type of algorithm is trial division? | {
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0.154517382... | Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | This motivates the concept of a problem being hard for a complexity class. A problem X is hard for a class of problems C if every problem in C can be reduced to X. Thus no problem in C is harder than X, since an algorithm for X allows us to solve any problem in C. Of course, the notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems. | deterministic | 96,633 |
57297d421d046914007794e9 | Prime_number | Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite. | When using a probabilistic algorithm, how is the probability that the number is composite expressed mathematically? | {
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-0.20104... | Packet mode communication may be implemented with or without intermediate forwarding nodes (packet switches or routers). Packets are normally forwarded by intermediate network nodes asynchronously using first-in, first-out buffering, but may be forwarded according to some scheduling discipline for fair queuing, traffic shaping, or for differentiated or guaranteed quality of service, such as weighted fair queuing or leaky bucket. In case of a shared physical medium (such as radio or 10BASE5), the packets may be delivered according to a multiple access scheme. | A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state. One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm. | A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. The input string for a computational problem is referred to as a problem instance, and should not be confused with the problem itself. In computational complexity theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing. The instance is a number (e.g. 15) and the solution is "yes" if the number is prime and "no" otherwise (in this case "no"). Stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input. | 1/(1-p)n | 96,634 |
57297ed93f37b3190047845f | Prime_number | A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number. | What is one straightforward case of a probabilistic test? | {
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57297ed93f37b31900478460 | Prime_number | A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number. | What does the Fermat primality test depend upon? | {
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0.099648125469684... | On October 25–28, 1990, Rukh held its second congress and declared that its principal goal was the "renewal of independent statehood for Ukraine". On October 28 UAOC faithful, supported by Ukrainian Catholics, demonstrated near St. Sophia’s Cathedral as newly elected Russian Orthodox Church Patriarch Aleksei and Metropolitan Filaret celebrated liturgy at the shrine. On November 1, the leaders of the Ukrainian Greek Catholic Church and of the Ukrainian Autocephalous Orthodox Church, respectively, Metropolitan Volodymyr Sterniuk and Patriarch Mstyslav, met in Lviv during anniversary commemorations of the 1918 proclamation of the Western Ukrainian National Republic. | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | np≡n (mod p) | 96,636 |
57297ed93f37b31900478461 | Prime_number | A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number. | What type of numbers demonstrate a flaw with the Fermat primality test? | {
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0.1121794655919075... | iPods cannot play music files from competing music stores that use rival-DRM technologies like Microsoft's protected WMA or RealNetworks' Helix DRM. Example stores include Napster and MSN Music. RealNetworks claims that Apple is creating problems for itself by using FairPlay to lock users into using the iTunes Store. Steve Jobs stated that Apple makes little profit from song sales, although Apple uses the store to promote iPod sales. However, iPods can also play music files from online stores that do not use DRM, such as eMusic or Amie Street. | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | composite numbers (the Carmichael numbers) | 96,637 |
57297ed93f37b31900478462 | Prime_number | A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number. | What is the name of one impressive continuation of the Fermat primality test? | {
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0.092218078672... | Wood has a long history of being used as fuel, which continues to this day, mostly in rural areas of the world. Hardwood is preferred over softwood because it creates less smoke and burns longer. Adding a woodstove or fireplace to a home is often felt to add ambiance and warmth. | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | Baillie-PSW | 96,638 |
57297ed93f37b31900478463 | Prime_number | A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number. | What is the name of another compelling continuation of the Fermat primality test? | {
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0.12848380208015442... | Michel Djotodia took over as president and in May 2013 Central African Republic's Prime Minister Nicolas Tiangaye requested a UN peacekeeping force from the UN Security Council and on 31 May former President Bozizé was indicted for crimes against humanity and incitement of genocide. The security situation did not improve during June–August 2013 and there were reports of over 200,000 internally displaced persons (IDPs) as well as human rights abuses and renewed fighting between Séléka and Bozizé supporters. | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | If a problem X is in C and hard for C, then X is said to be complete for C. This means that X is the hardest problem in C. (Since many problems could be equally hard, one might say that X is one of the hardest problems in C.) Thus the class of NP-complete problems contains the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. Because the problem P = NP is not solved, being able to reduce a known NP-complete problem, Π2, to another problem, Π1, would indicate that there is no known polynomial-time solution for Π1. This is because a polynomial-time solution to Π1 would yield a polynomial-time solution to Π2. Similarly, because all NP problems can be reduced to the set, finding an NP-complete problem that can be solved in polynomial time would mean that P = NP. | Solovay-Strassen tests | 96,639 |
572980f9af94a219006aa4d1 | Prime_number | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | Of what form are Sophie Germain primes? | {
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-0.1308034211397... | The first recorded settlement in what is now Newcastle was Pons Aelius, a Roman fort and bridge across the River Tyne. It was given the family name of the Roman Emperor Hadrian, who founded it in the 2nd century AD. This rare honour suggests that Hadrian may have visited the site and instituted the bridge on his tour of Britain. The population of Pons Aelius at this period was estimated at 2,000. Fragments of Hadrian's Wall are still visible in parts of Newcastle, particularly along the West Road. The course of the "Roman Wall" can be traced eastwards to the Segedunum Roman fort in Wallsend—the "wall's end"—and to the supply fort Arbeia in South Shields. The extent of Hadrian's Wall was 73 miles (117 km), spanning the width of Britain; the Wall incorporated the Vallum, a large rearward ditch with parallel mounds, and was constructed primarily for defence, to prevent unwanted immigration and the incursion of Pictish tribes from the north, not as a fighting line for a major invasion. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | 2p + 1 | 96,640 |
572980f9af94a219006aa4d2 | Prime_number | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | Of what form are Mersenne primes? | {
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0.21699234... | In a 2009 national readership survey The Times was found to have the highest number of ABC1 25–44 readers and the largest numbers of readers in London of any of the "quality" papers. | There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n. | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | 2p − 1 | 96,641 |
572980f9af94a219006aa4d3 | Prime_number | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | What test is especially useful for numbers of the form 2p - 1? | {
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-0.235025584697... | The Consortium first published The Unicode Standard (ISBN 0-321-18578-1) in 1991 and continues to develop standards based on that original work. The latest version of the standard, Unicode 8.0, was released in June 2015 and is available from the consortium's website. The last of the major versions (versions x.0) to be published in book form was Unicode 5.0 (ISBN 0-321-48091-0), but since Unicode 6.0 the full text of the standard is no longer being published in book form. In 2012, however, it was announced that only the core specification for Unicode version 6.1 would be made available as a 692-page print-on-demand paperback. Unlike the previous major version printings of the Standard, the print-on-demand core specification does not include any code charts or standard annexes, but the entire standard, including the core specification, will still remain freely available on the Unicode website. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | The Lucas–Lehmer test | 96,642 |
572980f9af94a219006aa4d4 | Prime_number | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | What is the name of one type of prime where p+1 or p-1 takes a certain shape? | {
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147,
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147
],
"text": [
"primorial primes",
"Fermat",
"Sophie Germain",
"Sophie Germain",
"Sophie Germain"
]
} | What is the name of [MASK] type of prime where p+1 or p-1 takes a certain shape? | [
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0.05181373655796051,
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0.1399217993... | The classical period of Greek civilization covers a time spanning from the early 5th century BC to the death of Alexander the Great, in 323 BC (some authors prefer to split this period into 'Classical', from the end of the Persian wars to the end of the Peloponnesian War, and 'Fourth Century', up to the death of Alexander). It is so named because it set the standards by which Greek civilization would be judged in later eras. The Classical period is also described as the "Golden Age" of Greek civilization, and its art, philosophy, architecture and literature would be instrumental in the formation and development of Western culture. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | primorial primes | 96,643 |
572980f9af94a219006aa4d5 | Prime_number | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | What is the name of another type of prime here p+1 or p-1 takes a certain shape? | {
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229,
247,
211,
211,
211
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"text": [
"Fermat primes",
"Mersenne",
"primorial primes",
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} | What is the name of another type of prime here p+1 or p-1 takes a certain shape? | [
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0.14676775038... | When the news arrived at Paris of the surrender at Sedan of Napoleon III and 80,000 men, the Second Empire was overthrown by a popular uprising in Paris, which forced the proclamation of a Provisional Government and a Third Republic by general Trochu, Favre and Gambetta at Paris on 4 September, the new government calling itself the Government of National Defence. After the German victory at Sedan, most of the French standing army was either besieged in Metz or prisoner of the Germans, who hoped for an armistice and an end to the war. Bismarck wanted an early peace but had difficulty in finding a legitimate French authority with which to negotiate. The Government of National Defence had no electoral mandate, the Emperor was a captive and the Empress in exile but there had been no abdication de jure and the army was still bound by an oath of allegiance to the defunct imperial régime. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | Fermat primes | 96,644 |
572982e66aef051400154f92 | Prime_number | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | What is the name of one type of computing method that is used to find prime numbers? | {
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118,
118,
118,
118,
118
],
"text": [
"distributed computing",
"distributed computing",
"distributed",
"distributed computing",
"distributed computing"
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} | What is the name of [MASK] type of computing method that is used to find prime numbers? | [
0.1325530707836151,
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0.1099496185779571... | In 1864, Andrew Johnson (a War Democrat from Tennessee) was elected Vice President under Abraham Lincoln. He became President after Lincoln's assassination in 1865. Under Johnson's lenient re-admission policy, Tennessee was the first of the seceding states to have its elected members readmitted to the U.S. Congress, on July 24, 1866. Because Tennessee had ratified the Fourteenth Amendment, it was the only one of the formerly secessionist states that did not have a military governor during the Reconstruction period. | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O(n2) for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O(n log n). The best case occurs when each pivoting divides the list in half, also needing O(n log n) time. | distributed computing | 96,645 |
572982e66aef051400154f93 | Prime_number | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | In what year was the Great Internet Mersenne Prime Search project conducted? | {
"answer_start": [
141,
144,
144,
144,
144
],
"text": [
"In 2009",
"2009",
"2009",
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} | In what year was the Great Internet Mersenne Prime Search project conducted? | [
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0.0898797512054... | Greeks have a long tradition of valuing and investing in paideia (education). Paideia was one of the highest societal values in the Greek and Hellenistic world while the first European institution described as a university was founded in 5th century Constantinople and operated in various incarnations until the city's fall to the Ottomans in 1453. The University of Constantinople was Christian Europe's first secular institution of higher learning since no theological subjects were taught, and considering the original meaning of the world university as a corporation of students, the world’s first university as well. | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | In 2009 | 96,646 |
572982e66aef051400154f94 | Prime_number | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | the Great Internet Mersenne Prime Search, what was the prize for finding a prime with at least 10 million digits? | {
"answer_start": [
213,
213,
213,
215,
213
],
"text": [
"US$100,000",
"US$100,000",
"US$100,000",
"$100,000",
"US$100,000"
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} | the Great Internet Mersenne Prime Search, what was the prize for finding a prime with [MASK] digits? | [
-0.25631147623062134,
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0.1442825347185135,
0.37027549743652344,
0.00955124944448471,
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0.14718592166900635,
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0.0514601357281... | Swaziland's currency is pegged to the South African Rand, subsuming Swaziland's monetary policy to South Africa. Customs duties from the Southern African Customs Union, which may equal as much as 70% of government revenue this year, and worker remittances from South Africa substantially supplement domestically earned income. Swaziland is not poor enough to merit an IMF program; however, the country is struggling to reduce the size of the civil service and control costs at public enterprises. The government is trying to improve the atmosphere for foreign direct investment. | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | US$100,000 | 96,647 |
572982e66aef051400154f95 | Prime_number | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | What organization offers monetary awards for identifying primes with at least 100 million digits? | {
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297,
297,
291,
340
],
"text": [
"The Electronic Frontier Foundation",
"Electronic Frontier Foundation",
"Electronic Frontier Foundation",
". The Electronic Frontier Foundation",
"$150,000"
]
} | What organization offers monetary awards for identifying primes with [MASK] digits? | [
0.05664552003145218,
0.19828812777996063,
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0.036671675741... | The traditional Charleston accent has long been noted in the state and throughout the South. It is typically heard in wealthy white families who trace their families back generations in the city. It has ingliding or monophthongal long mid-vowels, raises ay and aw in certain environments, and is nonrhotic. Sylvester Primer of the College of Charleston wrote about aspects of the local dialect in his late 19th-century works: "Charleston Provincialisms" (1887) and "The Huguenot Element in Charleston's Provincialisms", published in a German journal. He believed the accent was based on the English as it was spoken by the earliest settlers, therefore derived from Elizabethan England and preserved with modifications by Charleston speakers. The rapidly disappearing "Charleston accent" is still noted in the local pronunciation of the city's name. Some elderly (and usually upper-class) Charleston natives ignore the 'r' and elongate the first vowel, pronouncing the name as "Chah-l-ston". Some observers attribute these unique features of Charleston's speech to its early settlement by French Huguenots and Sephardic Jews (who were primarily English speakers from London), both of whom played influential roles in Charleston's early development and history.[citation needed] | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | The Electronic Frontier Foundation | 96,648 |
572982e76aef051400154f96 | Prime_number | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | In what interval are some of the greatest primes without a distinct form discovered in? | {
"answer_start": [
765,
765,
765,
765,
765
],
"text": [
"[256kn + 1, 256k(n + 1) − 1]",
"[256kn + 1, 256k(n + 1) − 1]",
"[256kn + 1, 256k(n + 1) − 1]",
"[256kn + 1, 256k(n + 1) − 1]",
"[256kn + 1, 256k(n + 1) − 1]."
]
} | In what interval are some of the greatest primes without a distinct form discovered in? | [
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0.329311... | NE1fm launched on 8 June 2007, the first full-time community radio station in the area. Newcastle Student Radio is run by students from both of the city's universities, broadcasting from Newcastle University's student's union building during term time. Radio Tyneside has been the voluntary hospital radio service for most hospitals across Newcastle and Gateshead since 1951, broadcasting on Hospedia and online. The city also has a Radio Lollipop station based at the Great North Children's Hospital in the Newcastle Royal Victoria Infirmary. | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | [256kn + 1, 256k(n + 1) − 1] | 96,649 |
572985011d04691400779501 | Prime_number | are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. | What is name of the function used for the largest integer not greater than the number in question? | {
"answer_start": [
53,
57,
57,
57,
57
],
"text": [
"the floor function",
"floor",
"floor",
"floor function",
"floor function"
]
} | What is name of the function used for the largest integer not greater than the number in question? | [
-0.20062270760536194,
-0.0016695187659934163,
-0.009956900961697102,
0.13743636012077332,
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0.42170456051826477,
-0.10454946756362915,
-0.12977340817451477,
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0.33848512172698975,
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0.17814530432224274,
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0.12581... | The situation in Switzerland and Liechtenstein is different from the rest of the German-speaking countries. The Swiss German dialects are the default everyday language in virtually every situation, whereas standard German is seldom spoken. Some Swiss German speakers perceive standard German to be a foreign language. | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | the floor function | 96,650 |
572985011d04691400779502 | Prime_number | are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. | Who first proved Bertrand's postulate? | {
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} | Who first proved [MASK] 's postulate? | [
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0.28227... | When not celebrating Mass but still serving a liturgical function, such as the semiannual Urbi et Orbi papal blessing, some Papal Masses and some events at Ecumenical Councils, cardinal deacons can be recognized by the dalmatics they would don with the simple white mitre (so called mitra simplex). | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | Chebyshev | 96,651 |
572985011d04691400779503 | Prime_number | are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. | For what size natural number does Bertrand's postulate hold? | {
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0.357598364... | In 1968 Ronald Melzack and Kenneth Casey described pain in terms of its three dimensions: "sensory-discriminative" (sense of the intensity, location, quality and duration of the pain), "affective-motivational" (unpleasantness and urge to escape the unpleasantness), and "cognitive-evaluative" (cognitions such as appraisal, cultural values, distraction and hypnotic suggestion). They theorized that pain intensity (the sensory discriminative dimension) and unpleasantness (the affective-motivational dimension) are not simply determined by the magnitude of the painful stimulus, but "higher" cognitive activities can influence perceived intensity and unpleasantness. Cognitive activities "may affect both sensory and affective experience or they may modify primarily the affective-motivational dimension. Thus, excitement in games or war appears to block both dimensions of pain, while suggestion and placebos may modulate the affective-motivational dimension and leave the sensory-discriminative dimension relatively undisturbed." (p. 432) The paper ends with a call to action: "Pain can be treated not only by trying to cut down the sensory input by anesthetic block, surgical intervention and the like, but also by influencing the motivational-affective and cognitive factors as well." (p. 435) | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. Such ramification questions occur even in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations | any natural number n > 3 | 96,652 |
572985011d04691400779504 | Prime_number | are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. | How is the prime number p in Bertrand's postulate expressed mathematically? | {
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"n < p < 2n − 2",
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} | How is the prime number p in [MASK] 's postulate expressed mathematically? | [
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0.238391935825... | After 539 Ravenna was reconquered by the Romans in the form of the Eastern Roman Empire (Byzantine Empire) and became the seat of the Exarchate of Ravenna. The greatest development of Christian mosaics unfolded in the second half of the 6th century. Outstanding examples of Byzantine mosaic art are the later phase mosaics in the Basilica of San Vitale and Basilica of Sant'Apollinare Nuovo. The mosaic depicting Emperor Saint Justinian I and Empress Theodora in the Basilica of San Vitale were executed shortly after the Byzantine conquest. The mosaics of the Basilica of Sant'Apollinare in Classe were made around 549. The anti-Arian theme is obvious in the apse mosaic of San Michele in Affricisco, executed in 545–547 (largely destroyed; the remains in Berlin). | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers Z[i], that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z[i], 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not. | n < p < 2n − 2 | 96,653 |
572985011d04691400779505 | Prime_number | are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. | On what theorem is the formula that frequently generates the number 2 and all other primes precisely once based on? | {
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459,
459
],
"text": [
"Wilson's theorem",
"Wilson's",
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} | On what theorem is the formula that frequently generates the number [MASK] and all other primes precisely once based on? | [
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0.31175526976... | Nuclear strike is the ability of nuclear forces to rapidly and accurately strike targets which the enemy holds dear in a devastating manner. If a crisis occurs, rapid generation and, if necessary, deployment of nuclear strike capabilities will demonstrate US resolve and may prompt an adversary to alter the course of action deemed threatening to our national interest. Should deterrence fail, the President may authorize a precise, tailored response to terminate the conflict at the lowest possible level and lead to a rapid cessation of hostilities. Post-conflict, regeneration of a credible nuclear deterrent capability will deter further aggression. The Air Force may present a credible force posture in either the Continental United States, within a theater of operations, or both to effectively deter the range of potential adversaries envisioned in the 21st century. This requires the ability to engage targets globally using a variety of methods; therefore, the Air Force should possess the ability to induct, train, assign, educate and exercise individuals and units to rapidly and effectively execute missions that support US NDO objectives. Finally, the Air Force regularly exercises and evaluates all aspects of nuclear operations to ensure high levels of performance. | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | Wilson's theorem | 96,654 |
572987e46aef051400154fa2 | Prime_number | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | What is another way to state the condition that infinitely many primes can exist only if a and q are coprime? | {
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"text": [
"their greatest common divisor is one",
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} | What is another way to state the condition that infinitely many primes can exist only if a and q are coprime? | [
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0.477817475795745... | There were 13 finalists this season, but two were eliminated in the first result show of the finals. A new feature introduced was the "Judges' Save", and Matt Giraud was saved from elimination at the top seven by the judges when he received the fewest votes. The next week, Lil Rounds and Anoop Desai were eliminated. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number. | their greatest common divisor is one | 96,655 |
572987e46aef051400154fa3 | Prime_number | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | If a and q are coprime, which theorem holds that an arithmetic progression has an infinite number of primes? | {
"answer_start": [
149,
149,
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149
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"Dirichlet's theorem",
"Dirichlet's",
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} | If a and q are coprime, which theorem holds that an arithmetic progression has an infinite number of primes? | [
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0.238011136... | The term middle east as a noun and adjective was common in the 19th century in nearly every context except diplomacy and archaeology. An uncountable number of places appear to have had their middle easts from gardens to regions, including the United States. The innovation of the term "Near East" to mean the holdings of the Ottoman Empire as early as the Crimean War had left a geographical gap. The East Indies, or "Far East," derived ultimately from Ptolemy's "India Beyond the Ganges." The Ottoman Empire ended at the eastern border of Iraq. "India This Side of the Ganges" and Iran had been omitted. The archaeologists counted Iran as "the Near East" because Old Persian cuneiform had been found there. This usage did not sit well with the diplomats; India was left in an equivocal state. They needed a regional term. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers. | Dirichlet's theorem | 96,656 |
572987e46aef051400154fa4 | Prime_number | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | What is the density of all primes compatible with a modulo 9? | {
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713,
713,
713,
713
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"text": [
"1/6",
"1/6",
"1/6",
"1/6"
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} | What is the density of all primes compatible with a modulo [MASK]? | [
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-0.0353791937... | The province's name derives from the Zhe River (浙江, Zhè Jiāng), the former name of the Qiantang River which flows past Hangzhou and whose mouth forms Hangzhou Bay. It is usually glossed as meaning "Crooked" or "Bent River", from the meaning of Chinese 折, but is more likely a phono-semantic compound formed from adding 氵 (the "water" radical used for river names) to phonetic 折 (pinyin zhé but reconstructed Old Chinese *tet), preserving a proto-Wu name of the local Yue, similar to Yuhang, Kuaiji, and Jiang. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | 1/6 | 96,657 |
572987e46aef051400154fa5 | Prime_number | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | If q=9 and a=3,6 or 9, how many primes would be in the progression? | {
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477,
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"text": [
"at most one prime number",
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} | If q=9 and [MASK] or [MASK], how many primes would be in the progression? | [
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0.28238952159... | In cases where the criminalized behavior is pure speech, civil disobedience can consist simply of engaging in the forbidden speech. An example would be WBAI's broadcasting the track "Filthy Words" from a George Carlin comedy album, which eventually led to the 1978 Supreme Court case of FCC v. Pacifica Foundation. Threatening government officials is another classic way of expressing defiance toward the government and unwillingness to stand for its policies. For example, Joseph Haas was arrested for allegedly sending an email to the Lebanon, New Hampshire city councilors stating, "Wise up or die." | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations". | at most one prime number | 96,658 |
572987e46aef051400154fa6 | Prime_number | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | If q=9 and a=1,2,4,5,7, or 8, how many primes would be in a progression? | {
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550,
550,
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550
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"text": [
"infinitely many prime numbers",
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} | If q=9 and [MASK], or [MASK], how many primes would be in a progression? | [
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0.29131156206130... | The most famous passage in Burke's Reflections was his description of the events of 5–6 October 1789 and the part of Marie-Antoinette in them. Burke's account differs little from modern historians who have used primary sources. His use of flowery language to describe it, however, provoked both praise and criticism. Philip Francis wrote to Burke saying that what he wrote of Marie-Antoinette was "pure foppery". Edward Gibbon, however, reacted differently: "I adore his chivalry". Burke was informed by an Englishman who had talked with the Duchesse de Biron, that when Marie-Antoinette was reading the passage, she burst into tears and took considerable time to finish reading it. Price had rejoiced that the French king had been "led in triumph" during the October Days, but to Burke this symbolised the opposing revolutionary sentiment of the Jacobins and the natural sentiments of those who shared his own view with horror—that the ungallant assault on Marie-Antoinette—was a cowardly attack on a defenceless woman. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols, | infinitely many prime numbers | 96,659 |
572989846aef051400154fc0 | Prime_number | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | What function is related to prime numbers? | {
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4,
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} | What function is related to prime numbers? | [
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0.23868262... | The main response of the immune system to tumors is to destroy the abnormal cells using killer T cells, sometimes with the assistance of helper T cells. Tumor antigens are presented on MHC class I molecules in a similar way to viral antigens. This allows killer T cells to recognize the tumor cell as abnormal. NK cells also kill tumorous cells in a similar way, especially if the tumor cells have fewer MHC class I molecules on their surface than normal; this is a common phenomenon with tumors. Sometimes antibodies are generated against tumor cells allowing for their destruction by the complement system. | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. | The zeta function | 96,660 |
572989846aef051400154fc1 | Prime_number | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | What type of value would the zeta function have if there were finite primes? | {
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235,
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"text": [
"a finite value",
"finite",
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} | What type of value would the zeta function have if there were finite primes? | [
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0.31698688... | Research into LCPS (low cost private schools) found that over 5 years to July 2013, debate around LCPSs to achieving Education for All (EFA) objectives was polarised and finding growing coverage in international policy. The polarisation was due to disputes around whether the schools are affordable for the poor, reach disadvantaged groups, provide quality education, support or undermine equality, and are financially sustainable. The report examined the main challenges encountered by development organisations which support LCPSs. Surveys suggest these types of schools are expanding across Africa and Asia. This success is attributed to excess demand. These surveys found concern for: | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | a finite value | 96,661 |
572989846aef051400154fc2 | Prime_number | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | What property of the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... shows that there is an infinite number of primes? | {
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304,
304,
320
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"text": [
"diverges",
"diverges",
"exceeds any given number"
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} | What property of the harmonic series [MASK] + 1/2 + [MASK] + 1/4 +... shows that there is an infinite number of primes? | [
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0.1744921356... | Neptune's mass of 1.0243×1026 kg, is intermediate between Earth and the larger gas giants: it is 17 times that of Earth but just 1/19th that of Jupiter.[d] Its gravity at 1 bar is 11.15 m/s2, 1.14 times the surface gravity of Earth, and surpassed only by Jupiter. Neptune's equatorial radius of 24,764 km is nearly four times that of Earth. Neptune, like Uranus, is an ice giant, a subclass of giant planet, due to their smaller size and higher concentrations of volatiles relative to Jupiter and Saturn. In the search for extrasolar planets, Neptune has been used as a metonym: discovered bodies of similar mass are often referred to as "Neptunes", just as scientists refer to various extrasolar bodies as "Jupiters". | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | diverges | 96,662 |
572989846aef051400154fc3 | Prime_number | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | What does it mean when a harmonic series diverges? | {
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320,
320,
320,
320
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"text": [
"exceeds any given number",
"exceeds any given number",
"exceeds any given number",
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} | What does it mean when a harmonic series diverges? | [
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0.24443167... | The use of drones by the Central Intelligence Agency in Pakistan to carry out operations associated with the Global War on Terror sparks debate over sovereignty and the laws of war. The U.S. Government uses the CIA rather than the U.S. Air Force for strikes in Pakistan in order to avoid breaching sovereignty through military invasion. The United States was criticized by[according to whom?] a report on drone warfare and aerial sovereignty for abusing the term 'Global War on Terror' to carry out military operations through government agencies without formally declaring war. | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | The most commonly used reduction is a polynomial-time reduction. This means that the reduction process takes polynomial time. For example, the problem of squaring an integer can be reduced to the problem of multiplying two integers. This means an algorithm for multiplying two integers can be used to square an integer. Indeed, this can be done by giving the same input to both inputs of the multiplication algorithm. Thus we see that squaring is not more difficult than multiplication, since squaring can be reduced to multiplication. | exceeds any given number | 96,663 |
572989846aef051400154fc4 | Prime_number | The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler, | Of what mathematical nature is the Basel problem? | {
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465,
458,
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"identity",
"algebraic",
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} | Of what mathematical nature is the Basel problem? | [
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-0.105654... | Paris and its close suburbs is home to numerous newspapers, magazines and publications including Le Monde, Le Figaro, Libération, Le Nouvel Observateur, Le Canard enchaîné, La Croix, Pariscope, Le Parisien (in Saint-Ouen), Les Échos, Paris Match (Neuilly-sur-Seine), Réseaux & Télécoms, Reuters France, and L'Officiel des Spectacles. France's two most prestigious newspapers, Le Monde and Le Figaro, are the centrepieces of the Parisian publishing industry. Agence France-Presse is France's oldest, and one of the world's oldest, continually operating news agencies. AFP, as it is colloquially abbreviated, maintains its headquarters in Paris, as it has since 1835. France 24 is a television news channel owned and operated by the French government, and is based in Paris. Another news agency is France Diplomatie, owned and operated by the Ministry of Foreign and European Affairs, and pertains solely to diplomatic news and occurrences. | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | Of course, some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following: | identity | 96,664 |
57298ef11d0469140077952d | Prime_number | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | When was the Riemann hypothesis proposed? | {
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45,
45,
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"1859",
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} | When was the [MASK] hypothesis proposed? | [
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0.27006... | Others have pointed out that there were not enough of these loans made to cause a crisis of this magnitude. In an article in Portfolio Magazine, Michael Lewis spoke with one trader who noted that "There weren’t enough Americans with [bad] credit taking out [bad loans] to satisfy investors' appetite for the end product." Essentially, investment banks and hedge funds used financial innovation to enable large wagers to be made, far beyond the actual value of the underlying mortgage loans, using derivatives called credit default swaps, collateralized debt obligations and synthetic CDOs. | Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh–Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh–Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture. | The theoretical difficulties arise from the fact that all of the methods except the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect. If these theories are slightly inaccurate – though there is no evidence at present to suggest they are – the methods would not give accurate values for the Planck constant. More importantly, the values of the Planck constant obtained in this way cannot be used as tests of the theories without falling into a circular argument. Fortunately, there are other statistical ways of testing the theories, and the theories have yet to be refuted. | 1859 | 96,665 |
57298ef11d0469140077952e | Prime_number | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | According to the Riemann hypothesis, all zeroes of the ζ-function have real part equal to 1/2 except for what values of s? | {
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74,
78,
78,
74
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"s = −2, −4, ...,",
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} | According to the [MASK] hypothesis, all zeroes of the ζ- function have real part equal to [MASK] except for what values of s? | [
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... | The catechism is one of Luther's most personal works. "Regarding the plan to collect my writings in volumes," he wrote, "I am quite cool and not at all eager about it because, roused by a Saturnian hunger, I would rather see them all devoured. For I acknowledge none of them to be really a book of mine, except perhaps the Bondage of the Will and the Catechism." The Small Catechism has earned a reputation as a model of clear religious teaching. It remains in use today, along with Luther's hymns and his translation of the Bible. | Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh–Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh–Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | s = −2, −4, ..., | 96,666 |
57298ef11d0469140077952f | Prime_number | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | What does the Riemann hypothesis state the source of irregularity in the distribution of points comes from? | {
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402,
402,
402,
402
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"text": [
"random noise",
"random noise",
"random noise",
"random noise"
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} | What does the Riemann hypothesis state the source of irregularity in the distribution of points comes from? | [
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0.4146246612071... | The Tucson metro area is served by many local television stations and is the 68th largest designated market area (DMA) in the U.S. with 433,310 homes (0.39% of the total U.S.). It is limited to the three counties of southeastern Arizona (Pima, Santa Cruz, and Cochise) The major television networks serving Tucson are: KVOA 4 (NBC), KGUN 9 (ABC), KMSB-TV 11 (Fox), KOLD-TV 13 (CBS), KTTU 18 (My Network TV) and KWBA 58 (The CW). KUAT-TV 6 is a PBS affiliate run by the University of Arizona (as is sister station KUAS 27). | Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh–Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh–Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture. | There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n. | random noise | 96,667 |
57298ef11d04691400779530 | Prime_number | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | What type of prime distribution does the Riemann hypothesis propose is also true for short intervals near X? | {
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0.263345688581... | Kadamba (345 – 525 CE) was an ancient royal dynasty of Karnataka, India that ruled northern Karnataka and the Konkan from Banavasi in present-day Uttara Kannada district. At the peak of their power under King Kakushtavarma, the Kadambas of Banavasi ruled large parts of modern Karnataka state. The dynasty was founded by Mayurasharma in 345 CE which at later times showed the potential of developing into imperial proportions, an indication to which is provided by the titles and epithets assumed by its rulers. King Mayurasharma defeated the armies of Pallavas of Kanchi possibly with help of some native tribes. The Kadamba fame reached its peak during the rule of Kakusthavarma, a notable ruler with whom even the kings of Gupta Dynasty of northern India cultivated marital alliances. The Kadambas were contemporaries of the Western Ganga Dynasty and together they formed the earliest native kingdoms to rule the land with absolute autonomy. The dynasty later continued to rule as a feudatory of larger Kannada empires, the Chalukya and the Rashtrakuta empires, for over five hundred years during which time they branched into minor dynasties known as the Kadambas of Goa, Kadambas of Halasi and Kadambas of Hangal. | Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh–Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh–Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture. | In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis. | asymptotic distribution | 96,668 |
57298ef11d04691400779531 | Prime_number | The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason. | What type of prime distribution is characterized about x/log x of numbers less than x? | {
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0.24952653050... | Most analyses place Kerry's voting record on the left within the Senate Democratic caucus. During the 2004 presidential election he was portrayed as a staunch liberal by conservative groups and the Bush campaign, who often noted that in 2003 Kerry was rated the National Journal's top Senate liberal. However, that rating was based only upon voting on legislation within that past year. In fact, in terms of career voting records, the National Journal found that Kerry is the 11th most liberal member of the Senate. Most analyses find that Kerry is at least slightly more liberal than the typical Democratic Senator. Kerry has stated that he opposes privatizing Social Security, supports abortion rights for adult women and minors, supports same-sex marriage, opposes capital punishment except for terrorists, supports most gun control laws, and is generally a supporter of trade agreements. Kerry supported the North American Free Trade Agreement and Most Favored Nation status for China, but opposed the Central American Free Trade Agreement.[citation needed] | Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable. The Rayleigh–Jeans law makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make Planck's law, which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves h in both the numerator and the denominator. The influence of h in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh–Jeans law could not be done by changing the values of h, of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture. | Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. | asymptotic distribution | 96,669 |
57299021af94a219006aa50c | Prime_number | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | Which conjecture holds that every even integer n greater than 2 can be expressed as a sum of two primes? | {
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0.3926760... | Prior to the Golden Age of Mandolins, France had a history with the mandolin, with mandolinists playing in Paris until the Napoleonic Wars. The players, teachers and composers included Giovanni Fouchetti, Eduardo Mezzacapo, Gabriele Leon, and Gervasio. During the Golden age itself (1880s-1920s), the mandolin had a strong presence in France. Prominent mandolin players or composers included Jules Cottin and his sister Madeleine Cottin, Jean Pietrapertosa, and Edgar Bara. Paris had dozens of "estudiantina" mandolin orchestras in the early 1900s. Mandolin magazines included L'Estudiantina, Le Plectre, École de la mandolie. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | Goldbach's conjecture | 96,670 |
57299021af94a219006aa50b | Prime_number | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | When did Landau propose his four conjectural problems? | {
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0.02886504493653... | Three main parades take place during Carnival. The first is held on the first day, during which the "Carnival King" (either a person in costume or an effigy) rides through the city on his carriage. The second is held on the first Sunday of the festival and the participants are mainly children. The third and largest takes place on the last day of Carnival and involves hundreds of people walking in costume along the town's longest avenue. The latter two parades are open to anyone who wishes to participate. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | Von Neumann's famous 9-page paper started life as a talk at Princeton and then became a paper in Germany, which was eventually translated into English. His interest in economics that led to that paper began as follows: When lecturing at Berlin in 1928 and 1929 he spent his summers back home in Budapest, and so did the economist Nicholas Kaldor, and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann found some faults in that book and corrected them, for example, replacing equations by inequalities. He noticed that Walras's General Equilibrium Theory and Walras' Law, which led to systems of simultaneous linear equations, could produce the absurd result that the profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced the paper. | 1912 | 96,671 |
57299021af94a219006aa50d | Prime_number | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | As of February 2011, how many numbers has Goldbach's conjecture been proven to? | {
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0.4471205770969... | During the Republic, any person who wished to hold public office had to conform to the Reformed Church and take an oath to this effect. The extent to which different religions or denominations were persecuted depended much on the time period and regional or city leaders. In the beginning, this was especially focused on Roman Catholics, being the religion of the enemy. In 17th-century Leiden, for instance, people opening their homes to services could be fined 200 guilders (a year's wage for a skilled tradesman) and banned from the city. Throughout this, however, personal freedom of religion existed and was one factor – along with economic reasons – in causing large immigration of religious refugees from other parts of Europe. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite. | all numbers up to n = 2 · 1017 | 96,672 |
57299021af94a219006aa50e | Prime_number | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | Which theorem states that all large odd integers can be expressed as a sum of three primes? | {
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0.314389944... | Behind the scenes, the bookers in a company will place the title on the most accomplished performer, or those the bookers believe will generate fan interest in terms of event attendance and television viewership. Lower ranked titles may also be used on the performers who show potential, thus allowing them greater exposure to the audience. However other circumstances may also determine the use of a championship. A combination of a championship's lineage, the caliber of performers as champion, and the frequency and manner of title changes, dictates the audience's perception of the title's quality, significance and reputation. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | Vinogradov's theorem | 96,673 |
57299021af94a219006aa50f | Prime_number | In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011[update], this conjecture has been verified for all numbers up to n = 2 · 1017. Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | Which theorem states that every large even integer can be written as a prime summed with a semiprime? | {
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0.27386200... | In addition to basic uniform clothing, various badges are used by the USAF to indicate a billet assignment or qualification-level for a given assignment. Badges can also be used as merit-based or service-based awards. Over time, various badges have been discontinued and are no longer distributed. Authorized badges include the Shields of USAF Fire Protection, and Security Forces, and the Missile Badge (or "pocket rocket"), which is earned after working in a missile system maintenance or missile operations capacity for at least one year. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | The integer factorization problem is the computational problem of determining the prime factorization of a given integer. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. No efficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographic systems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP and co-UP). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP will equal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takes time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantum algorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say much about where the problem lies with respect to non-quantum complexity classes. | Chen's theorem | 96,674 |
572991943f37b319004784a1 | Prime_number | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | What conjecture holds that there is an infinite amount of twin primes? | {
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0.35285112261772... | In many ways, the Paleocene continued processes that had begun during the late Cretaceous Period. During the Paleocene, the continents continued to drift toward their present positions. Supercontinent Laurasia had not yet separated into three continents. Europe and Greenland were still connected. North America and Asia were still intermittently joined by a land bridge, while Greenland and North America were beginning to separate. The Laramide orogeny of the late Cretaceous continued to uplift the Rocky Mountains in the American west, which ended in the succeeding epoch. South and North America remained separated by equatorial seas (they joined during the Neogene); the components of the former southern supercontinent Gondwana continued to split apart, with Africa, South America, Antarctica and Australia pulling away from each other. Africa was heading north toward Europe, slowly closing the Tethys Ocean, and India began its migration to Asia that would lead to a tectonic collision and the formation of the Himalayas. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January 2016[update], the largest known prime number has 22,338,618 decimal digits. | twin prime conjecture | 96,675 |
572991943f37b319004784a2 | Prime_number | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | What is a twin prime? | {
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0.133992731... | Guinea-Bissau was once part of the kingdom of Gabu, as well as part of the Mali Empire. Parts of this kingdom persisted until the 18th century, while a few others were under some rule by the Portuguese Empire since the 16th century. In the 19th century, it was colonized as Portuguese Guinea. Upon independence, declared in 1973 and recognised in 1974, the name of its capital, Bissau, was added to the country's name to prevent confusion with Guinea (formerly French Guinea). Guinea-Bissau has a history of political instability since independence, and no elected president has successfully served a full five-year term. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | pairs of primes with difference 2 | 96,676 |
572991943f37b319004784a3 | Prime_number | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | Which conjecture holds that for any positive integer n, there is an infinite amount of pairs of consecutive primes differing by 2n? | {
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} | Which conjecture holds that for any positive integer n, there is an infinite amount of pairs of consecutive primes differing by [MASK]? | [
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0.34497866034... | The Austrian engineer Paul Eisler invented the printed circuit as part of a radio set while working in England around 1936. Around 1943 the USA began to use the technology on a large scale to make proximity fuses for use in World War II. After the war, in 1948, the USA released the invention for commercial use. Printed circuits did not become commonplace in consumer electronics until the mid-1950s, after the Auto-Sembly process was developed by the United States Army. At around the same time in Britain work along similar lines was carried out by Geoffrey Dummer, then at the RRDE. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes. | Polignac's conjecture | 96,677 |
572991943f37b319004784a4 | Prime_number | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | Of what form is the infinite amount of primes that comprise the special cases of Schinzel's hypothesis? | {
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439,
439,
439,
439
],
"text": [
"n2 + 1",
"n2 + 1",
"n2 + 1.",
"n2 + 1"
]
} | Of what form is the infinite amount of primes that comprise the special cases of [MASK] 's hypothesis? | [
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0.3836275935173034... | The doctrines of the Assumption or Dormition of Mary relate to her death and bodily assumption to Heaven. The Roman Catholic Church has dogmaically defined the doctrine of the Assumption, which was done in 1950 by Pope Pius XII in Munificentissimus Deus. Whether the Virgin Mary died or not is not defined dogmatically, however, although a reference to the death of Mary are made in Munificentissimus Deus. In the Eastern Orthodox Church, the Assumption of the Virgin Mary is believed, and celebrated with her Dormition, where they believe she died. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann. | n2 + 1 | 96,678 |
572991943f37b319004784a5 | Prime_number | A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n. It is conjectured there are infinitely many primes of the form n2 + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture. | What conjecture holds that there are always a minimum of 4 primes between the squares of consecutive primes greater than 2? | {
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521,
521,
521,
521
],
"text": [
"Brocard's conjecture",
"Brocard's",
"Brocard's conjecture",
"Brocard's"
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} | What conjecture holds that there are always a minimum of [MASK] primes between the squares of consecutive primes greater than [MASK]? | [
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0.145329564809... | Before printing was widely adopted in the 19th century, the Quran was transmitted in manuscripts made by calligraphers and copyists. The earliest manuscripts were written in Ḥijāzī-type script. The Hijazi style manuscripts nevertheless confirm that transmission of the Quran in writing began at an early stage. Probably in the ninth century, scripts began to feature thicker strokes, which are traditionally known as Kufic scripts. Toward the end of the ninth century, new scripts began to appear in copies of the Quran and replace earlier scripts. The reason for discontinuation in the use of the earlier style was that it took too long to produce and the demand for copies was increasing. Copyists would therefore chose simpler writing styles. Beginning in the 11th century, the styles of writing employed were primarily the naskh, muhaqqaq, rayḥānī and, on rarer occasions, the thuluth script. Naskh was in very widespread use. In North Africa and Spain, the Maghribī style was popular. More distinct is the Bihari script which was used solely in the north of India. Nastaʻlīq style was also rarely used in Persian world. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | are prime for any natural number n. Here represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. | Brocard's conjecture | 96,679 |
57299326af94a219006aa515 | Prime_number | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | Besides the study of prime numbers, what general theory was considered the official example of pure mathematics? | {
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17,
17,
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17
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"number theory",
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} | Besides the study of prime numbers, what general theory was considered the official example of pure mathematics? | [
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0.052698828279972... | Further distinctions in self-concept, called "differentiation," occur as the adolescent recognizes the contextual influences on their own behavior and the perceptions of others, and begin to qualify their traits when asked to describe themselves. Differentiation appears fully developed by mid-adolescence. Peaking in the 7th-9th grades, the personality traits adolescents use to describe themselves refer to specific contexts, and therefore may contradict one another. The recognition of inconsistent content in the self-concept is a common source of distress in these years (see Cognitive dissonance), but this distress may benefit adolescents by encouraging structural development. | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | The 1990s, along with a rise in object-oriented programming, saw a growth in how data in various databases were handled. Programmers and designers began to treat the data in their databases as objects. That is to say that if a person's data were in a database, that person's attributes, such as their address, phone number, and age, were now considered to belong to that person instead of being extraneous data. This allows for relations between data to be relations to objects and their attributes and not to individual fields. The term "object-relational impedance mismatch" described the inconvenience of translating between programmed objects and database tables. Object databases and object-relational databases attempt to solve this problem by providing an object-oriented language (sometimes as extensions to SQL) that programmers can use as alternative to purely relational SQL. On the programming side, libraries known as object-relational mappings (ORMs) attempt to solve the same problem. | number theory | 96,680 |
57299326af94a219006aa516 | Prime_number | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | What British mathematician took pride in doing work that he felt had no military benefit? | {
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360,
360,
360,
360
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"G. H. Hardy",
"G. H. Hardy",
"G. H. Hardy",
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} | What [MASK] mathematician took pride in doing work that he felt had no military benefit? | [
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-0.188468366... | Nobles were born into a noble family, adopted by a noble family (this was abolished in 1633) or ennobled by a king or Sejm for various reasons (bravery in combat, service to the state, etc.—yet this was the rarest means of gaining noble status). Many nobles were, in actuality, really usurpers, being commoners, who moved into another part of the country and falsely pretended to noble status. Hundreds of such false nobles were denounced by Hieronim Nekanda Trepka in his Liber generationis plebeanorium (or Liber chamorum) in the first half of the 16th century. The law forbade non-nobles from owning nobility-estates and promised the estate to the denouncer. Trepka was an impoverished nobleman who lived a townsman life and collected hundreds of such stories hoping to take over any of such estates. It does not seem he ever succeeded in proving one at the court. Many sejms issued decrees over the centuries in an attempt to resolve this issue, but with little success. It is unknown what percentage of the Polish nobility came from the 'lower' orders of society, but most historians agree that nobles of such base origins formed a 'significant' element of the szlachta. | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | In the extreme empiricism of the neopositivists—at least before the 1930s—any genuinely synthetic assertion must be reducible to an ultimate assertion (or set of ultimate assertions) that expresses direct observations or perceptions. In later years, Carnap and Neurath abandoned this sort of phenomenalism in favor of a rational reconstruction of knowledge into the language of an objective spatio-temporal physics. That is, instead of translating sentences about physical objects into sense-data, such sentences were to be translated into so-called protocol sentences, for example, "X at location Y and at time T observes such and such." The central theses of logical positivism (verificationism, the analytic-synthetic distinction, reductionism, etc.) came under sharp attack after World War II by thinkers such as Nelson Goodman, W.V. Quine, Hilary Putnam, Karl Popper, and Richard Rorty. By the late 1960s, it had become evident to most philosophers that the movement had pretty much run its course, though its influence is still significant among contemporary analytic philosophers such as Michael Dummett and other anti-realists. | G. H. Hardy | 96,681 |
57299326af94a219006aa517 | Prime_number | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | When was it discovered that prime numbers could applied to the creation of public key cryptography algorithms? | {
"answer_start": [
488,
492,
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492
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"text": [
"the 1970s",
"1970s",
"1970s",
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} | When was it discovered that prime numbers could applied to the creation of public key cryptography algorithms? | [
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0.02389821596... | Some Normans joined Turkish forces to aid in the destruction of the Armenians vassal-states of Sassoun and Taron in far eastern Anatolia. Later, many took up service with the Armenian state further south in Cilicia and the Taurus Mountains. A Norman named Oursel led a force of "Franks" into the upper Euphrates valley in northern Syria. From 1073 to 1074, 8,000 of the 20,000 troops of the Armenian general Philaretus Brachamius were Normans—formerly of Oursel—led by Raimbaud. They even lent their ethnicity to the name of their castle: Afranji, meaning "Franks." The known trade between Amalfi and Antioch and between Bari and Tarsus may be related to the presence of Italo-Normans in those cities while Amalfi and Bari were under Norman rule in Italy. | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | A Turing machine is a mathematical model of a general computing machine. It is a theoretical device that manipulates symbols contained on a strip of tape. Turing machines are not intended as a practical computing technology, but rather as a thought experiment representing a computing machine—anything from an advanced supercomputer to a mathematician with a pencil and paper. It is believed that if a problem can be solved by an algorithm, there exists a Turing machine that solves the problem. Indeed, this is the statement of the Church–Turing thesis. Furthermore, it is known that everything that can be computed on other models of computation known to us today, such as a RAM machine, Conway's Game of Life, cellular automata or any programming language can be computed on a Turing machine. Since Turing machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing machine is the most commonly used model in complexity theory. | the 1970s | 96,682 |
57299326af94a219006aa518 | Prime_number | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | Besides public key cryptography, what is another application for prime numbers? | {
"answer_start": [
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664,
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664
],
"text": [
"hash tables",
"hash tables",
"hash tables and pseudorandom number generators",
"hash tables and pseudorandom number generators"
]
} | Besides public key cryptography, what is another application for prime numbers? | [
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0.00256870547... | The modern literary language is usually considered to date from the time of Alexander Pushkin (Алекса́ндр Пу́шкин) in the first third of the 19th century. Pushkin revolutionized Russian literature by rejecting archaic grammar and vocabulary (so-called "высо́кий стиль" — "high style") in favor of grammar and vocabulary found in the spoken language of the time. Even modern readers of younger age may only experience slight difficulties understanding some words in Pushkin's texts, since relatively few words used by Pushkin have become archaic or changed meaning. In fact, many expressions used by Russian writers of the early 19th century, in particular Pushkin, Mikhail Lermontov (Михаи́л Ле́рмонтов), Nikolai Gogol (Никола́й Го́голь), Aleksander Griboyedov (Алекса́ндр Грибое́дов), became proverbs or sayings which can be frequently found even in modern Russian colloquial speech. | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | In most computers, individual instructions are stored as machine code with each instruction being given a unique number (its operation code or opcode for short). The command to add two numbers together would have one opcode; the command to multiply them would have a different opcode, and so on. The simplest computers are able to perform any of a handful of different instructions; the more complex computers have several hundred to choose from, each with a unique numerical code. Since the computer's memory is able to store numbers, it can also store the instruction codes. This leads to the important fact that entire programs (which are just lists of these instructions) can be represented as lists of numbers and can themselves be manipulated inside the computer in the same way as numeric data. The fundamental concept of storing programs in the computer's memory alongside the data they operate on is the crux of the von Neumann, or stored program[citation needed], architecture. In some cases, a computer might store some or all of its program in memory that is kept separate from the data it operates on. This is called the Harvard architecture after the Harvard Mark I computer. Modern von Neumann computers display some traits of the Harvard architecture in their designs, such as in CPU caches. | hash tables | 96,683 |
57299326af94a219006aa519 | Prime_number | For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance. However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators. | What type of number generators make use of prime numbers? | {
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0.11911... | Victoria wrote to her uncle Leopold, whom Victoria considered her "best and kindest adviser", to thank him "for the prospect of great happiness you have contributed to give me, in the person of dear Albert ... He possesses every quality that could be desired to render me perfectly happy. He is so sensible, so kind, and so good, and so amiable too. He has besides the most pleasing and delightful exterior and appearance you can possibly see." However at 17, Victoria, though interested in Albert, was not yet ready to marry. The parties did not undertake a formal engagement, but assumed that the match would take place in due time. | There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way. | The 1990s, along with a rise in object-oriented programming, saw a growth in how data in various databases were handled. Programmers and designers began to treat the data in their databases as objects. That is to say that if a person's data were in a database, that person's attributes, such as their address, phone number, and age, were now considered to belong to that person instead of being extraneous data. This allows for relations between data to be relations to objects and their attributes and not to individual fields. The term "object-relational impedance mismatch" described the inconvenience of translating between programmed objects and database tables. Object databases and object-relational databases attempt to solve this problem by providing an object-oriented language (sometimes as extensions to SQL) that programmers can use as alternative to purely relational SQL. On the programming side, libraries known as object-relational mappings (ORMs) attempt to solve the same problem. | pseudorandom number generators | 96,684 |
572995d46aef051400154fe8 | Prime_number | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | Assuming p is a prime other than 2 or 5, then, according to Fermat's theorem, what type of decimal will 1/p always be? | {
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0.072483763098... | Population has outstripped the supply of freshwater, usually from rainfall. The northern atolls get 50 inches (1,300 mm) of rainfall annually; the southern atolls about twice that. The threat of drought is commonplace throughout the island chains. | But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP. | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | a recurring decimal | 96,685 |
572995d46aef051400154fe9 | Prime_number | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | According to Fermat's theorem, what period does 1/p always have assuming p is prime that is not 2 or 5? | {
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0.1428319066762... | As Ashkenazi Jews moved away from Europe, mostly in the form of aliyah to Israel, or immigration to North America, and other English-speaking areas; and Europe (particularly France) and Latin America, the geographic isolation that gave rise to Ashkenazim has given way to mixing with other cultures, and with non-Ashkenazi Jews who, similarly, are no longer isolated in distinct geographic locales. Hebrew has replaced Yiddish as the primary Jewish language for many Ashkenazi Jews, although many Hasidic and Hareidi groups continue to use Yiddish in daily life. (There are numerous Ashkenazi Jewish anglophones and Russian-speakers as well, although English and Russian are not originally Jewish languages.) | But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP. | along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann. | p − 1 | 96,686 |
572995d46aef051400154fea | Prime_number | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | According to Wilson's theorem, what factorial must be divisible by p if some integer p > 1 is to be considered prime? | {
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0.1783515512... | Currently, the rapid influx of northerners and immigrants from Latin America is steadily increasing ethnic and religious diversity: the number of Roman Catholics and Jews in the state has increased, as well as general religious diversity. The second-largest Protestant denomination in North Carolina after Baptist traditions is Methodism, which is strong in the northern Piedmont, especially in populous Guilford County. There are also a substantial number of Quakers in Guilford County and northeastern North Carolina. Many universities and colleges in the state have been founded on religious traditions, and some currently maintain that affiliation, including: | But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP. | The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed] | (p − 1)! + 1 | 96,687 |
572995d46aef051400154feb | Prime_number | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | According to Wilson's theorem, what factorial must be divisible by n if some integer n > 4 is to be considered composite? | {
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0.15787181258... | The Greek islands are traditionally grouped into the following clusters: The Argo-Saronic Islands in the Saronic gulf near Athens, the Cyclades, a large but dense collection occupying the central part of the Aegean Sea, the North Aegean islands, a loose grouping off the west coast of Turkey, the Dodecanese, another loose collection in the southeast between Crete and Turkey, the Sporades, a small tight group off the coast of northeast Euboea, and the Ionian Islands, located to the west of the mainland in the Ionian Sea. | But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP. | After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor. | (n − 1)! | 96,688 |
572995d46aef051400154fec | Prime_number | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | What condition what must be satisfied in order for 1/p to be expressed in base q instead of base 10 and still have a period of p - 1? | {
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0.173632726073265... | Under the terms of the concluding Treaty of Versailles signed in 1919, the empire reached its greatest extent with the addition of 1,800,000 square miles (4,700,000 km2) and 13 million new subjects. The colonies of Germany and the Ottoman Empire were distributed to the Allied powers as League of Nations mandates. Britain gained control of Palestine, Transjordan, Iraq, parts of Cameroon and Togo, and Tanganyika. The Dominions themselves also acquired mandates of their own: the Union of South Africa gained South-West Africa (modern-day Namibia), Australia gained German New Guinea, and New Zealand Western Samoa. Nauru was made a combined mandate of Britain and the two Pacific Dominions. | But bounding the computation time above by some concrete function f(n) often yields complexity classes that depend on the chosen machine model. For instance, the language {xx | x is any binary string} can be solved in linear time on a multi-tape Turing machine, but necessarily requires quadratic time in the model of single-tape Turing machines. If we allow polynomial variations in running time, Cobham-Edmonds thesis states that "the time complexities in any two reasonable and general models of computation are polynomially related" (Goldreich 2008, Chapter 1.2). This forms the basis for the complexity class P, which is the set of decision problems solvable by a deterministic Turing machine within polynomial time. The corresponding set of function problems is FP. | The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January 2016[update], the largest known prime number has 22,338,618 decimal digits. | p is not a prime factor of q | 96,689 |
572996c73f37b319004784b3 | Prime_number | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | What is one type of public key cryptography algorithm? | {
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0.0127334706485... | The highest point in the state is Clingmans Dome at 6,643 feet (2,025 m). Clingmans Dome, which lies on Tennessee's eastern border, is the highest point on the Appalachian Trail, and is the third highest peak in the United States east of the Mississippi River. The state line between Tennessee and North Carolina crosses the summit. The state's lowest point is the Mississippi River at the Mississippi state line (the lowest point in Memphis, nearby, is at 195 ft (59 m)). The geographical center of the state is located in Murfreesboro. | In a further refinement of the direct use of probabilistic modelling, statistical estimates can be coupled to an algorithm called arithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a finite-state machine to produce a string of encoded bits from a series of input data symbols. It can achieve superior compression to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of the probability distribution of the input data. An early example of the use of arithmetic coding was its use as an optional (but not widely used) feature of the JPEG image coding standard. It has since been applied in various other designs including H.264/MPEG-4 AVC and HEVC for video coding. | In almost all modern computers, each memory cell is set up to store binary numbers in groups of eight bits (called a byte). Each byte is able to represent 256 different numbers (28 = 256); either from 0 to 255 or −128 to +127. To store larger numbers, several consecutive bytes may be used (typically, two, four or eight). When negative numbers are required, they are usually stored in two's complement notation. Other arrangements are possible, but are usually not seen outside of specialized applications or historical contexts. A computer can store any kind of information in memory if it can be represented numerically. Modern computers have billions or even trillions of bytes of memory. | RSA | 96,690 |
572996c73f37b319004784b4 | Prime_number | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | What is another type of public key cryptography algorithm? | {
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0.00297358... | Comcast Corporation, formerly registered as Comcast Holdings,[note 1] is an American multinational mass media company and is the largest broadcasting and largest cable company in the world by revenue. It is the second largest pay-TV company after the AT&T-DirecTV acquisition, largest cable TV company and largest home Internet service provider in the United States, and the nation's third largest home telephone service provider. Comcast services U.S. residential and commercial customers in 40 states and the District of Columbia. The company's headquarters are located in Philadelphia, Pennsylvania. | In a further refinement of the direct use of probabilistic modelling, statistical estimates can be coupled to an algorithm called arithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a finite-state machine to produce a string of encoded bits from a series of input data symbols. It can achieve superior compression to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of the probability distribution of the input data. An early example of the use of arithmetic coding was its use as an optional (but not widely used) feature of the JPEG image coding standard. It has since been applied in various other designs including H.264/MPEG-4 AVC and HEVC for video coding. | Most browsers support HTTP Secure and offer quick and easy ways to delete the web cache, download history, form and search history, cookies, and browsing history. For a comparison of the current security vulnerabilities of browsers, see comparison of web browsers. | the Diffie–Hellman key exchange | 96,691 |
572996c73f37b319004784b5 | Prime_number | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | How many bits are often in the primes used for RSA public key cryptography algorithms? | {
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} | How many bits are often in the primes used for RSA public key cryptography algorithms? | [
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0.0714469403... | In the centre of Basel, the first major city in the course of the stream, is located the "Rhine knee"; this is a major bend, where the overall direction of the Rhine changes from West to North. Here the High Rhine ends. Legally, the Central Bridge is the boundary between High and Upper Rhine. The river now flows North as Upper Rhine through the Upper Rhine Plain, which is about 300 km long and up to 40 km wide. The most important tributaries in this area are the Ill below of Strasbourg, the Neckar in Mannheim and the Main across from Mainz. In Mainz, the Rhine leaves the Upper Rhine Valley and flows through the Mainz Basin. | In a further refinement of the direct use of probabilistic modelling, statistical estimates can be coupled to an algorithm called arithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a finite-state machine to produce a string of encoded bits from a series of input data symbols. It can achieve superior compression to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of the probability distribution of the input data. An early example of the use of arithmetic coding was its use as an optional (but not widely used) feature of the JPEG image coding standard. It has since been applied in various other designs including H.264/MPEG-4 AVC and HEVC for video coding. | The committee debated the possibility of a shift function (like in ITA2), which would allow more than 64 codes to be represented by a six-bit code. In a shifted code, some character codes determine choices between options for the following character codes. It allows compact encoding, but is less reliable for data transmission as an error in transmitting the shift code typically makes a long part of the transmission unreadable. The standards committee decided against shifting, and so ASCII required at least a seven-bit code.:215, 236 § 4 | 512-bit | 96,692 |
572996c73f37b319004784b6 | Prime_number | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | On what type of exponentiation does the Diffie–Hellman key exchange depend on? | {
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} | On what type of exponentiation does the Diffie–Hellman key exchange depend on? | [
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0.26669082045555... | Birds have been domesticated by humans both as pets and for practical purposes. Colourful birds, such as parrots and mynas, are bred in captivity or kept as pets, a practice that has led to the illegal trafficking of some endangered species. Falcons and cormorants have long been used for hunting and fishing, respectively. Messenger pigeons, used since at least 1 AD, remained important as recently as World War II. Today, such activities are more common either as hobbies, for entertainment and tourism, or for sports such as pigeon racing. | In a further refinement of the direct use of probabilistic modelling, statistical estimates can be coupled to an algorithm called arithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a finite-state machine to produce a string of encoded bits from a series of input data symbols. It can achieve superior compression to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of the probability distribution of the input data. An early example of the use of arithmetic coding was its use as an optional (but not widely used) feature of the JPEG image coding standard. It has since been applied in various other designs including H.264/MPEG-4 AVC and HEVC for video coding. | are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers. | modular exponentiation | 96,693 |
572996c73f37b319004784b7 | Prime_number | Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example, 512-bit primes are frequently used for RSA and 1024-bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem. | How many bits are typically used in the primes for the Diffie–Hellman key exchange? | {
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"1024-bit",
"1024",
"1024",
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} | How many bits are typically used in the primes for the Diffie–Hellman key exchange? | [
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0.184767112135... | One of John's principal challenges was acquiring the large sums of money needed for his proposed campaigns to reclaim Normandy. The Angevin kings had three main sources of income available to them, namely revenue from their personal lands, or demesne; money raised through their rights as a feudal lord; and revenue from taxation. Revenue from the royal demesne was inflexible and had been diminishing slowly since the Norman conquest. Matters were not helped by Richard's sale of many royal properties in 1189, and taxation played a much smaller role in royal income than in later centuries. English kings had widespread feudal rights which could be used to generate income, including the scutage system, in which feudal military service was avoided by a cash payment to the king. He derived income from fines, court fees and the sale of charters and other privileges. John intensified his efforts to maximise all possible sources of income, to the extent that he has been described as "avaricious, miserly, extortionate and moneyminded". John also used revenue generation as a way of exerting political control over the barons: debts owed to the crown by the king's favoured supporters might be forgiven; collection of those owed by enemies was more stringently enforced. | In a further refinement of the direct use of probabilistic modelling, statistical estimates can be coupled to an algorithm called arithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a finite-state machine to produce a string of encoded bits from a series of input data symbols. It can achieve superior compression to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of the probability distribution of the input data. An early example of the use of arithmetic coding was its use as an optional (but not widely used) feature of the JPEG image coding standard. It has since been applied in various other designs including H.264/MPEG-4 AVC and HEVC for video coding. | There is active research to make computers out of many promising new types of technology, such as optical computers, DNA computers, neural computers, and quantum computers. Most computers are universal, and are able to calculate any computable function, and are limited only by their memory capacity and operating speed. However different designs of computers can give very different performance for particular problems; for example quantum computers can potentially break some modern encryption algorithms (by quantum factoring) very quickly. | 1024-bit | 96,694 |
572998673f37b319004784d5 | Prime_number | The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. | What type of insect employs the use of prime numbers in its evolutionary strategy? | {
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} | What type of insect employs the use of prime numbers in its evolutionary strategy? | [
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-0.03774906322... | John's campaign started well. In November John retook Rochester Castle from rebel baron William d'Aubigny in a sophisticated assault. One chronicler had not seen "a siege so hard pressed or so strongly resisted", whilst historian Reginald Brown describes it as "one of the greatest [siege] operations in England up to that time". Having regained the south-east John split his forces, sending William Longespée to retake the north side of London and East Anglia, whilst John himself headed north via Nottingham to attack the estates of the northern barons. Both operations were successful and the majority of the remaining rebels were pinned down in London. In January 1216 John marched against Alexander II of Scotland, who had allied himself with the rebel cause. John took back Alexander's possessions in northern England in a rapid campaign and pushed up towards Edinburgh over a ten-day period. | The act of predation can be broken down into a maximum of four stages: Detection of prey, attack, capture and finally consumption. The relationship between predator and prey is one that is typically beneficial to the predator, and detrimental to the prey species. Sometimes, however, predation has indirect benefits to the prey species, though the individuals preyed upon themselves do not benefit. This means that, at each applicable stage, predator and prey species are in an evolutionary arms race to maximize their respective abilities to obtain food or avoid being eaten. This interaction has resulted in a vast array of adaptations in both groups. | Unlike in higher animals, where parthenogenesis is rare, asexual reproduction may occur in plants by several different mechanisms. The formation of stem tubers in potato is one example. Particularly in arctic or alpine habitats, where opportunities for fertilisation of flowers by animals are rare, plantlets or bulbs, may develop instead of flowers, replacing sexual reproduction with asexual reproduction and giving rise to clonal populations genetically identical to the parent. This is one of several types of apomixis that occur in plants. Apomixis can also happen in a seed, producing a seed that contains an embryo genetically identical to the parent. | cicadas | 96,695 |
572998673f37b319004784d6 | Prime_number | The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. | Where do cicadas spend the majority of their lives? | {
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} | Where do cicadas spend the majority of their lives? | [
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-0.0509836040437... | On December 25, 1991, the Russian SFSR was renamed the Russian Federation. On December 26, 1991, the USSR was self-dissolved by the Soviet of Nationalities, which by that time was the only functioning house of the Supreme Soviet (the other house, Soviet of the Union, had already lost the quorum after recall of its members by the union republics). After dissolution of the USSR, Russia declared that it assumed the rights and obligations of the dissolved central Soviet government, including UN membership. | The act of predation can be broken down into a maximum of four stages: Detection of prey, attack, capture and finally consumption. The relationship between predator and prey is one that is typically beneficial to the predator, and detrimental to the prey species. Sometimes, however, predation has indirect benefits to the prey species, though the individuals preyed upon themselves do not benefit. This means that, at each applicable stage, predator and prey species are in an evolutionary arms race to maximize their respective abilities to obtain food or avoid being eaten. This interaction has resulted in a vast array of adaptations in both groups. | Many long-distance migrants appear to be genetically programmed to respond to changing day length. Species that move short distances, however, may not need such a timing mechanism, instead moving in response to local weather conditions. Thus mountain and moorland breeders, such as wallcreeper Tichodroma muraria and white-throated dipper Cinclus cinclus, may move only altitudinally to escape the cold higher ground. Other species such as merlin Falco columbarius and Eurasian skylark Alauda arvensis move further, to the coast or towards the south. Species like the chaffinch are much less migratory in Britain than those of continental Europe, mostly not moving more than 5 km in their lives. | as grubs underground | 96,696 |
572998673f37b319004784d7 | Prime_number | The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. | Other than 7 and 13, what other year interval do cicadas pupate? | {
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} | Other than [MASK] and [MASK], what other year interval do cicadas pupate? | [
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0.0909103602170... | In the early 1990s, Dell sold its products through Best Buy, Costco and Sam's Club stores in the United States. Dell stopped this practice in 1994, citing low profit-margins on the business, exclusively distributing through a direct-sales model for the next decade. In 2003, Dell briefly sold products in Sears stores in the U.S. In 2007, Dell started shipping its products to major retailers in the U.S. once again, starting with Sam's Club and Wal-Mart. Staples, the largest office-supply retailer in the U.S., and Best Buy, the largest electronics retailer in the U.S., became Dell retail partners later that same year. | The act of predation can be broken down into a maximum of four stages: Detection of prey, attack, capture and finally consumption. The relationship between predator and prey is one that is typically beneficial to the predator, and detrimental to the prey species. Sometimes, however, predation has indirect benefits to the prey species, though the individuals preyed upon themselves do not benefit. This means that, at each applicable stage, predator and prey species are in an evolutionary arms race to maximize their respective abilities to obtain food or avoid being eaten. This interaction has resulted in a vast array of adaptations in both groups. | However, the lifecycles of most living polychaetes, which are almost all marine animals, are unknown, and only about 25% of the 300+ species whose lifecycles are known follow this pattern. About 14% use a similar external fertilization but produce yolk-rich eggs, which reduce the time the larva needs to spend among the plankton, or eggs from which miniature adults emerge rather than larvae. The rest care for the fertilized eggs until they hatch – some by producing jelly-covered masses of eggs which they tend, some by attaching the eggs to their bodies and a few species by keeping the eggs within their bodies until they hatch. These species use a variety of methods for sperm transfer; for example, in some the females collect sperm released into the water, while in others the males have a penis that inject sperm into the female. There is no guarantee that this is a representative sample of polychaetes' reproductive patterns, and it simply reflects scientists' current knowledge. | 17 years | 96,697 |
572998673f37b319004784d8 | Prime_number | The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. | What is the logic behind the cicadas prime number evolutionary strategy? | {
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} | What is the logic behind the cicadas prime number evolutionary strategy? | [
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0.20156034... | In late February, large public rallies took place in Kiev to protest the election laws, on the eve of the March 26 elections to the USSR Congress of People's Deputies, and to call for the resignation of the first secretary of the Communist Party of Ukraine, Volodymyr Scherbytsky, lampooned as "the mastodon of stagnation." The demonstrations coincided with a visit to Ukraine by Soviet President Gorbachev. On February 26, 1989, between 20,000 and 30,000 people participated in an unsanctioned ecumenical memorial service in Lviv, marking the anniversary of the death of 19th Century Ukrainian artist and nationalist Taras Shevchenko. | The act of predation can be broken down into a maximum of four stages: Detection of prey, attack, capture and finally consumption. The relationship between predator and prey is one that is typically beneficial to the predator, and detrimental to the prey species. Sometimes, however, predation has indirect benefits to the prey species, though the individuals preyed upon themselves do not benefit. This means that, at each applicable stage, predator and prey species are in an evolutionary arms race to maximize their respective abilities to obtain food or avoid being eaten. This interaction has resulted in a vast array of adaptations in both groups. | The primary physiological cue for migration are the changes in the day length. These changes are also related to hormonal changes in the birds. In the period before migration, many birds display higher activity or Zugunruhe (German: migratory restlessness), first described by Johann Friedrich Naumann in 1795, as well as physiological changes such as increased fat deposition. The occurrence of Zugunruhe even in cage-raised birds with no environmental cues (e.g. shortening of day and falling temperature) has pointed to the role of circannual endogenous programs in controlling bird migrations. Caged birds display a preferential flight direction that corresponds with the migratory direction they would take in nature, changing their preferential direction at roughly the same time their wild conspecifics change course. | make it very difficult for predators to evolve that could specialize as predators | 96,698 |
572998673f37b319004784d9 | Prime_number | The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. | How much larger would cicada predator populations be if cicada outbreaks occurred at 14 and 15 year intervals? | {
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0.018775044009... | Southampton used to be home to a number of ferry services to the continent, with destinations such as San Sebastian, Lisbon, Tangier and Casablanca. A ferry port was built during the 1960s. However, a number of these relocated to Portsmouth and by 1996, there were no longer any car ferries operating from Southampton with the exception of services to the Isle of Wight. The land used for Southampton Ferry Port was sold off and a retail and housing development was built on the site. The Princess Alexandra Dock was converted into a marina. Reception areas for new cars now fill the Eastern Docks where passengers, dry docks and trains used to be. | The act of predation can be broken down into a maximum of four stages: Detection of prey, attack, capture and finally consumption. The relationship between predator and prey is one that is typically beneficial to the predator, and detrimental to the prey species. Sometimes, however, predation has indirect benefits to the prey species, though the individuals preyed upon themselves do not benefit. This means that, at each applicable stage, predator and prey species are in an evolutionary arms race to maximize their respective abilities to obtain food or avoid being eaten. This interaction has resulted in a vast array of adaptations in both groups. | Ninety-five percent of bird species are socially monogamous. These species pair for at least the length of the breeding season or—in some cases—for several years or until the death of one mate. Monogamy allows for both paternal care and biparental care, which is especially important for species in which females require males' assistance for successful brood-rearing. Among many socially monogamous species, extra-pair copulation (infidelity) is common. Such behaviour typically occurs between dominant males and females paired with subordinate males, but may also be the result of forced copulation in ducks and other anatids. Female birds have sperm storage mechanisms that allow sperm from males to remain viable long after copulation, a hundred days in some species. Sperm from multiple males may compete through this mechanism. For females, possible benefits of extra-pair copulation include getting better genes for her offspring and insuring against the possibility of infertility in her mate. Males of species that engage in extra-pair copulations will closely guard their mates to ensure the parentage of the offspring that they raise. | up to 2% higher | 96,699 |
57299a6f6aef051400155016 | Prime_number | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | What does the word prime generally suggest? | {
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0.186007186770439... | Between AD 300 and 1300 in the northern part of the state along the wide, fertile valley on the San Miguel River the Casas Grandes (Big Houses) culture developed into an advanced civilization. The Casas Grandes civilization is part of a major prehistoric archaeological culture known as Mogollon which is related to the Ancestral Pueblo culture. Paquime was the center of the Casas Grandes civilization. Extensive archaeological evidence shows commerce, agriculture, and hunting at Paquime and Cuarenta Casas (Forty Houses). | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | for any constant c. Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize the above sketched connection of fields and groups to infinite field extensions. An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group. | indecomposability | 96,700 |
57299a6f6aef051400155017 | Prime_number | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | For a field F containing 0 and 1, what would be the prime field? | {
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0.25887417793... | Red is the color at the end of the spectrum of visible light next to orange and opposite violet. Red color has a predominant light wavelength of roughly 620–740 nanometres. Red is one of the additive primary colors of visible light, along with green and blue, which in Red Green Blue (RGB) color systems are combined to create all the colors on a computer monitor or television screen. Red is also one of the subtractive primary colors, along with yellow and blue, of the RYB color space and traditional color wheel used by painters and artists. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | Of course, some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following: | the smallest subfield | 96,701 |
57299a6f6aef051400155019 | Prime_number | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | How can any knot be distinctively indicated? | {
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0.19924204051494... | According to the Framework Law (3549/2007), Public higher education "Highest Educational Institutions" (Ανώτατα Εκπαιδευτικά Ιδρύματα, Anótata Ekpaideytiká Idrýmata, "ΑΕΙ") consists of two parallel sectors:the University sector (Universities, Polytechnics, Fine Arts Schools, the Open University) and the Technological sector (Technological Education Institutions (TEI) and the School of Pedagogic and Technological Education). There are also State Non-University Tertiary Institutes offering vocationally oriented courses of shorter duration (2 to 3 years) which operate under the authority of other Ministries. Students are admitted to these Institutes according to their performance at national level examinations taking place after completion of the third grade of Lykeio. Additionally, students over twenty-two years old may be admitted to the Hellenic Open University through a form of lottery. The Capodistrian University of Athens is the oldest university in the eastern Mediterranean. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers. | as a connected sum of prime knots | 96,702 |
57299a6f6aef05140015501a | Prime_number | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | What is an additional meaning intended when the word prime is used? | {
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0.2437537759542465... | During 1824–28 Chopin spent his vacations away from Warsaw, at a number of locales.[n 4] In 1824 and 1825, at Szafarnia, he was a guest of Dominik Dziewanowski, the father of a schoolmate. Here for the first time he encountered Polish rural folk music. His letters home from Szafarnia (to which he gave the title "The Szafarnia Courier"), written in a very modern and lively Polish, amused his family with their spoofing of the Warsaw newspapers and demonstrated the youngster's literary gift. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | any object can be, essentially uniquely, decomposed into its prime components | 96,703 |
57299a6f6aef051400155018 | Prime_number | The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. Prime models and prime 3-manifolds are other examples of this type. | What does it mean for a knot to be considered indecomposable? | {
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0.1412274092... | Originating as the Jama'at al-Tawhid wal-Jihad in 1999, it pledged allegiance to al-Qaeda in 2004, participated in the Iraqi insurgency that followed the March 2003 invasion of Iraq by Western forces, joined the fight in the Syrian Civil War beginning in March 2011, and was expelled from al-Qaeda in early 2014, (which complained of its failure to consult and "notorious intransigence"). The group gained prominence after it drove Iraqi government forces out of key cities in western Iraq in a 2014 offensive. The group is adept at social media, posting Internet videos of beheadings of soldiers, civilians, journalists and aid workers, and is known for its destruction of cultural heritage sites. The United Nations has held ISIL responsible for human rights abuses and war crimes, and Amnesty International has reported ethnic cleansing by the group on a "historic scale". The group has been designated a terrorist organisation by the United Nations, the European Union and member states, the United States, India, Indonesia, Turkey, Saudi Arabia, Syria and other countries. | can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6. | Many known complexity classes are suspected to be unequal, but this has not been proved. For instance P ⊆ NP ⊆ PP ⊆ PSPACE, but it is possible that P = PSPACE. If P is not equal to NP, then P is not equal to PSPACE either. Since there are many known complexity classes between P and PSPACE, such as RP, BPP, PP, BQP, MA, PH, etc., it is possible that all these complexity classes collapse to one class. Proving that any of these classes are unequal would be a major breakthrough in complexity theory. | it cannot be written as the knot sum of two nontrivial knots | 96,704 |
57299c2c6aef051400155020 | Prime_number | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | What is the name of an algebraic structure in which addition, subtraction and multiplication are defined? | {
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0.088252186... | Heitaro Nakajima, who developed an early digital audio recorder within Japan's national public broadcasting organization NHK in 1970, became general manager of Sony's audio department in 1971. His team developed a digital PCM adaptor audio tape recorder using a Betamax video recorder in 1973. After this, in 1974 the leap to storing digital audio on an optical disc was easily made. Sony first publicly demonstrated an optical digital audio disc in September 1976. A year later, in September 1977, Sony showed the press a 30 cm disc that could play 60 minutes of digital audio (44,100 Hz sampling rate and 16-bit resolution) using MFM modulation. In September 1978, the company demonstrated an optical digital audio disc with a 150-minute playing time, 44,056 Hz sampling rate, 16-bit linear resolution, and cross-interleaved error correction code—specifications similar to those later settled upon for the standard Compact Disc format in 1980. Technical details of Sony's digital audio disc were presented during the 62nd AES Convention, held on 13–16 March 1979, in Brussels. Sony's AES technical paper was published on 1 March 1979. A week later, on 8 March, Philips publicly demonstrated a prototype of an optical digital audio disc at a press conference called "Philips Introduce Compact Disc" in Eindhoven, Netherlands. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January 2016[update], the largest known prime number has 22,338,618 decimal digits. | commutative ring R | 96,705 |
57299c2c6aef051400155021 | Prime_number | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | What is one general concept that applies to elements of commutative rings? | {
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0.08301630616188... | London's most popular sport is football and it has fourteen League football clubs, including five in the Premier League: Arsenal, Chelsea, Crystal Palace, Tottenham Hotspur, and West Ham United. Among other professional teams based in London include Fulham, Queens Park Rangers, Millwall and Charlton Athletic. In May 2012, Chelsea became the first London club to win the UEFA Champions League. Aside from Arsenal, Chelsea and Tottenham, none of the other London clubs have ever won the national league title. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | In particular, this norm gets smaller when a number is multiplied by p, in sharp contrast to the usual absolute value (also referred to as the infinite prime). While completing Q (roughly, filling the gaps) with respect to the absolute value yields the field of real numbers, completing with respect to the p-adic norm |−|p yields the field of p-adic numbers. These are essentially all possible ways to complete Q, by Ostrowski's theorem. Certain arithmetic questions related to Q or more general global fields may be transferred back and forth to the completed (or local) fields. This local-global principle again underlines the importance of primes to number theory. | prime elements | 96,706 |
57299c2c6aef051400155022 | Prime_number | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | What is another general concept that applies to elements of commutative rings? | {
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0.0653865039348602... | The Flushing Remonstrance shows support for separation of church and state as early as the mid-17th century, stating their opposition to religious persecution of any sort: "The law of love, peace and liberty in the states extending to Jews, Turks and Egyptians, as they are considered sons of Adam, which is the glory of the outward state of Holland, so love, peace and liberty, extending to all in Christ Jesus, condemns hatred, war and bondage." The document was signed December 27, 1657 by a group of English citizens in America who were affronted by persecution of Quakers and the religious policies of the Governor of New Netherland, Peter Stuyvesant. Stuyvesant had formally banned all religions other than the Dutch Reformed Church from being practiced in the colony, in accordance with the laws of the Dutch Republic. The signers indicated their "desire therefore in this case not to judge lest we be judged, neither to condemn least we be condemned, but rather let every man stand or fall to his own Master." Stuyvesant fined the petitioners and threw them in prison until they recanted. However, John Bowne allowed the Quakers to meet in his home. Bowne was arrested, jailed, and sent to the Netherlands for trial; the Dutch court exonerated Bowne. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | In particular, this norm gets smaller when a number is multiplied by p, in sharp contrast to the usual absolute value (also referred to as the infinite prime). While completing Q (roughly, filling the gaps) with respect to the absolute value yields the field of real numbers, completing with respect to the p-adic norm |−|p yields the field of p-adic numbers. These are essentially all possible ways to complete Q, by Ostrowski's theorem. Certain arithmetic questions related to Q or more general global fields may be transferred back and forth to the completed (or local) fields. This local-global principle again underlines the importance of primes to number theory. | irreducible elements | 96,707 |
57299c2c6aef051400155023 | Prime_number | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | What is one condition that an element p of R must satisfy in order to be considered a prime element? | {
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"it is neither zero nor a unit",
"neither zero nor a unit",
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} | What is [MASK] condition that an element p of R must satisfy in order to be considered a prime element? | [
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0.08584963530302048... | Luther spoke out against the Jews in Saxony, Brandenburg, and Silesia. Josel of Rosheim, the Jewish spokesman who tried to help the Jews of Saxony in 1537, later blamed their plight on "that priest whose name was Martin Luther—may his body and soul be bound up in hell!—who wrote and issued many heretical books in which he said that whoever would help the Jews was doomed to perdition." Josel asked the city of Strasbourg to forbid the sale of Luther's anti-Jewish works: they refused initially, but did so when a Lutheran pastor in Hochfelden used a sermon to urge his parishioners to murder Jews. Luther's influence persisted after his death. Throughout the 1580s, riots led to the expulsion of Jews from several German Lutheran states. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n. | it is neither zero nor a unit | 96,708 |
57299c2c6aef051400155024 | Prime_number | Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is | Under what condition is an element irreducible? | {
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"it is not a unit and cannot be written as a product of two ring elements that are not units"
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} | Under what condition is an element irreducible? | [
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0.1517297774553... | The unlicensed clone market has flourished following Nintendo's discontinuation of the NES. Some of the more exotic of these resulting systems have gone beyond the functionality of the original hardware and have included variations such as a portable system with a color LCD (e.g. PocketFami). Others have been produced with certain specialized markets in mind, such as an NES clone that functions as a rather primitive personal computer, which includes a keyboard and basic word processing software. These unauthorized clones have been helped by the invention of the so-called NES-on-a-chip. | The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. | In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g−1h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.d[›] | cannot be written as a product of two ring elements that are not units | 96,709 |
57299d1c1d04691400779581 | Prime_number | The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers Z[i], that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z[i], 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not. | What theorem remains valid in unique factorization domains? | {
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0.31144091486930... | With the emergence of ISIL and its capture of large areas of Iraq and Syria, a number of crises resulted that sparked international attention. ISIL had perpetrated sectarian killings and war crimes in both Iraq and Syria. Gains made in the Iraq war were rolled back as Iraqi army units abandoned their posts. Cities were taken over by the terrorist group which enforced its brand of Sharia law. The kidnapping and decapitation of numerous Western journalists and aid-workers also garnered interest and outrage among Western powers. The US intervened with airstrikes in Iraq over ISIL held territories and assets in August, and in September a coalition of US and Middle Eastern powers initiated a bombing campaign in Syria aimed at degrading and destroying ISIL and Al-Nusra-held territory. | Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Zp. Groups of order p2 can also be shown to be abelian, a statement which does not generalize to order p3, as the non-abelian group D4 of order 8 = 23 above shows. Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups. Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena. | Similarly, it is not known if L (the set of all problems that can be solved in logarithmic space) is strictly contained in P or equal to P. Again, there are many complexity classes between the two, such as NL and NC, and it is not known if they are distinct or equal classes. | The fundamental theorem of arithmetic | 96,710 |
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