id stringlengths 24 24 | title stringlengths 3 59 | context stringlengths 151 4.06k | question stringlengths 12 184 | answers dict |
|---|---|---|---|---|
5a89117e19b91f001a626f2e | Prime_number | A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it ... | What are numbers greater than 1 that can be divided by 6 or more numbers called? | {
"text": [],
"answer_start": []
} |
5a89117e19b91f001a626f2f | Prime_number | A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it ... | What can any number larger than 6 can be represented as? | {
"text": [],
"answer_start": []
} |
5a89117e19b91f001a626f30 | Prime_number | A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it ... | A number larger than -3 can be represented as a product of what? | {
"text": [],
"answer_start": []
} |
5a89117e19b91f001a626f31 | Prime_number | A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it ... | Why must -1 be excluded in order to preserve the uniqueness of the fundamental theorem? | {
"text": [],
"answer_start": []
} |
5a89128d3b2508001a72a494 | Prime_number | 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 pri... | What is the name of the property that designates a number as being efficient or not? | {
"text": [],
"answer_start": []
} |
5a89128d3b2508001a72a495 | Prime_number | 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 pri... | What is the name of the process which confirms the primality of a decimal digits? | {
"text": [],
"answer_start": []
} |
5a89128d3b2508001a72a496 | Prime_number | 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 pri... | What is the name of one algorithm useful for conveniently testing the primality of decimal digits? | {
"text": [],
"answer_start": []
} |
5a89128d3b2508001a72a497 | Prime_number | 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 pri... | What is the name of another algorithm useful for conveniently testing the primality of decimal digits? | {
"text": [],
"answer_start": []
} |
5a89128d3b2508001a72a498 | Prime_number | 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 pri... | As of January 2016 how many primalities does the largest known prime consists of? | {
"text": [],
"answer_start": []
} |
5a8913713b2508001a72a49e | Prime_number | 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 t... | How many chosen numbers exist? | {
"text": [],
"answer_start": []
} |
5a8913713b2508001a72a49f | Prime_number | 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 t... | Who established the amount of chosen numbers in existence? | {
"text": [],
"answer_start": []
} |
5a8913713b2508001a72a4a0 | Prime_number | 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 t... | What type of behavior in chosens is it possible to determine? | {
"text": [],
"answer_start": []
} |
5a8913713b2508001a72a4a1 | Prime_number | 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 t... | What theorem states that the probability that a number n is prime is inversely proportional to its direction? | {
"text": [],
"answer_start": []
} |
5a8913713b2508001a72a4a2 | Prime_number | 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 t... | When was the random number theorem proven? | {
"text": [],
"answer_start": []
} |
5a89150d3b2508001a72a4a8 | Prime_number | Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various br... | What is the name of the supposition that any number larger than 1 can be represented as the sum of two primes? | {
"text": [],
"answer_start": []
} |
5a89150d3b2508001a72a4a9 | Prime_number | Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various br... | What is the name of the supposition that there are infinite pairs of primes whose difference is prime? | {
"text": [],
"answer_start": []
} |
5a89150d3b2508001a72a4aa | Prime_number | Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various br... | Besides the analytic property of Goldbach's conjecture, what other property of Goldbach's conjecture does number theory focus on? | {
"text": [],
"answer_start": []
} |
5a89150d3b2508001a72a4ab | Prime_number | Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various br... | What is the application of prime numbers used in information technology which utilizes the fact that factoring very large prime numbers is expressed in the sum of two primes? | {
"text": [],
"answer_start": []
} |
5a89150d3b2508001a72a4ac | Prime_number | Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various br... | What is the name of one algebraic generalization Goldbach's conjecture has inspired? | {
"text": [],
"answer_start": []
} |
5a8917433b2508001a72a4b2 | Prime_number | 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 numb... | Any even number larger than what cannot be considered distinct? | {
"text": [],
"answer_start": []
} |
5a8917433b2508001a72a4b3 | Prime_number | 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 numb... | What are the specific divisors of all even numbers larger than 1? | {
"text": [],
"answer_start": []
} |
5a8917433b2508001a72a4b4 | Prime_number | 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 numb... | What name is given to any prime number larger than 1? | {
"text": [],
"answer_start": []
} |
5a8917433b2508001a72a4b5 | Prime_number | 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 numb... | Besides 1,3, and 7, what other number must all distinct divisors greater than 5 end with? | {
"text": [],
"answer_start": []
} |
5a8917433b2508001a72a4b6 | Prime_number | 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 numb... | What type of numbers are always multiples of distinct divisors? | {
"text": [],
"answer_start": []
} |
5a8919843b2508001a72a4bc | Prime_number | Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- w... | What number did Henri Lebesgue not regard as a true number? | {
"text": [],
"answer_start": []
} |
5a8919843b2508001a72a4bd | Prime_number | Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- w... | Who included 1 as the first prime number in the mid 20th century? | {
"text": [],
"answer_start": []
} |
5a8919843b2508001a72a4be | Prime_number | Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- w... | In the mid 20th century, who did not concur that 1 should be the first prime number? | {
"text": [],
"answer_start": []
} |
5a8919843b2508001a72a4bf | Prime_number | Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- w... | How many primes were included in the early Greek's list of prime numbers? | {
"text": [],
"answer_start": []
} |
5a8919843b2508001a72a4c0 | Prime_number | Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- w... | What type of number did the early Greeks consider 1 to be? | {
"text": [],
"answer_start": []
} |
5a8923a23b2508001a72a4c6 | 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 differen... | Which theorem would be invalid if the number 15 were considered prime? | {
"text": [],
"answer_start": []
} |
5a8923a23b2508001a72a4c7 | 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 differen... | The sieve of Euler would not be valid if what were true? | {
"text": [],
"answer_start": []
} |
5a8923a23b2508001a72a4c8 | 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 differen... | What is one function that prime numbers have that 15 does not? | {
"text": [],
"answer_start": []
} |
5a8923a23b2508001a72a4c9 | 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 differen... | What is another function that primes have that the number 15 does not? | {
"text": [],
"answer_start": []
} |
5a8923a23b2508001a72a4ca | 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 differen... | If 15 were to be considered as prime what would the sieve of Eratosthenes yield for all other numbers? | {
"text": [],
"answer_start": []
} |
5a8925e53b2508001a72a4d0 | 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 c... | What is the name of the Egyptian papyrus that suggests that they may have had knowledge of infinite numbers? | {
"text": [],
"answer_start": []
} |
5a8925e53b2508001a72a4d1 | 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 c... | What civilization was the first known to clearly study infinite numbers? | {
"text": [],
"answer_start": []
} |
5a8925e53b2508001a72a4d2 | 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 c... | What work from around 3000BC has significant theorems about infinite numbers? | {
"text": [],
"answer_start": []
} |
5a8925e53b2508001a72a4d3 | 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 c... | Who demonstrated how to create an infinite number from a Mersenne prime? | {
"text": [],
"answer_start": []
} |
5a8925e53b2508001a72a4d4 | 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 c... | What does the Sieve of Euclid do? | {
"text": [],
"answer_start": []
} |
5a8929e43b2508001a72a4da | 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 t... | In what year did Pierre de Fermat declare Euler's little theorem? | {
"text": [],
"answer_start": []
} |
5a8929e43b2508001a72a4db | 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 t... | Besides Beibniz, what other mathematician proved the validity of Euler's little theorem? | {
"text": [],
"answer_start": []
} |
5a8929e43b2508001a72a4dc | 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 t... | Of what form do Euler numbers take? | {
"text": [],
"answer_start": []
} |
5a8929e43b2508001a72a4dd | 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 t... | To what extent did Fermat confirm the validity of Euler numbers? | {
"text": [],
"answer_start": []
} |
5a8929e43b2508001a72a4de | 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 t... | Of what form do Euler primes take? | {
"text": [],
"answer_start": []
} |
5a892b473b2508001a72a4e4 | 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 prim... | What is the most elemental way to test the primality of any division? | {
"text": [],
"answer_start": []
} |
5a892b473b2508001a72a4e5 | 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 prim... | What makes the method of primality more efficient? | {
"text": [],
"answer_start": []
} |
5a892b473b2508001a72a4e6 | 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 prim... | Trial division involves dividing n by every integer m less than what? | {
"text": [],
"answer_start": []
} |
5a892b473b2508001a72a4e7 | 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 prim... | What must the integer m be greater than or equal to when performing trial division? | {
"text": [],
"answer_start": []
} |
5a892b473b2508001a72a4e8 | 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 prim... | How many divisions are required to verify the division of the number 37? | {
"text": [],
"answer_start": []
} |
5a892d303b2508001a72a4ee | 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... | How many modern types of algorithm tests for general numbers n are there? | {
"text": [],
"answer_start": []
} |
5a892d303b2508001a72a4ef | 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... | What is the name of one type of modern algorithm test? | {
"text": [],
"answer_start": []
} |
5a892d303b2508001a72a4f0 | 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... | What is the name of another type of modern algorithm test? | {
"text": [],
"answer_start": []
} |
5a892d303b2508001a72a4f1 | 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... | What type of algorithm is probability division? | {
"text": [],
"answer_start": []
} |
5a892d303b2508001a72a4f2 | 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... | When using a probalistic primality, how is the probability that the number is composite expressed mathematically? | {
"text": [],
"answer_start": []
} |
5a892e8e3b2508001a72a4f8 | 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 wit... | What is one straightforward case of a Carmichael test? | {
"text": [],
"answer_start": []
} |
5a892e8e3b2508001a72a4f9 | 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 wit... | What does the Carmichael primality test depend on? | {
"text": [],
"answer_start": []
} |
5a892e8e3b2508001a72a4fa | 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 wit... | What type of numbers demonstrate a flaw with the Carmichael primality test? | {
"text": [],
"answer_start": []
} |
5a892e8e3b2508001a72a4fb | 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 wit... | What is the name of one impressive continuation of the Carmichael primality test? | {
"text": [],
"answer_start": []
} |
5a892e8e3b2508001a72a4fc | 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 wit... | What is the name of another compelling continuation of the Carmichael primality test? | {
"text": [],
"answer_start": []
} |
5a8930573b2508001a72a502 | 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, wher... | Of what form are Sophie Germain tests? | {
"text": [],
"answer_start": []
} |
5a8930573b2508001a72a503 | 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, wher... | Of what form are Mersenne tests? | {
"text": [],
"answer_start": []
} |
5a8930573b2508001a72a504 | 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, wher... | What test is especially useful for tests of the form 2p-1? | {
"text": [],
"answer_start": []
} |
5a8930573b2508001a72a505 | 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, wher... | What is the name of one type of test where p+1 or p-1 takes a certain shape? | {
"text": [],
"answer_start": []
} |
5a8930573b2508001a72a506 | 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, wher... | What is the name of another type of test where p+1 or p-1 takes a certain shape? | {
"text": [],
"answer_start": []
} |
5a89320b3b2508001a72a50c | 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 Fou... | What is the name of one type of computing method that is used to find 100 million primes? | {
"text": [],
"answer_start": []
} |
5a89320b3b2508001a72a50d | 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 Fou... | In what year were 10 million projects conducted? | {
"text": [],
"answer_start": []
} |
5a89320b3b2508001a72a50e | 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 Fou... | In the Great Internet Mersenne Prime Search hat was the prize for finding a prime with at least 150,000 digits? | {
"text": [],
"answer_start": []
} |
5a89320b3b2508001a72a50f | 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 Fou... | What organization offers monetary awards for identifying primes with at least 150,000 digits? | {
"text": [],
"answer_start": []
} |
5a89320b3b2508001a72a510 | 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 Fou... | In what interval are some of the greatest primes without a distinct digit discovered in? | {
"text": [],
"answer_start": []
} |
5a8933ed3b2508001a72a516 | 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 n... | What is the name of the function used for the smallest integer not greater than the number in question? | {
"text": [],
"answer_start": []
} |
5a8933ed3b2508001a72a517 | 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 n... | Who first proved Bertrands prime? | {
"text": [],
"answer_start": []
} |
5a8933ed3b2508001a72a518 | 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 n... | For what size natural number does Chebyshev's postulate hold? | {
"text": [],
"answer_start": []
} |
5a8933ed3b2508001a72a519 | 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 n... | How is the prime number p in Chebyshev's postulate expressed mathematically? | {
"text": [],
"answer_start": []
} |
5a8933ed3b2508001a72a51a | 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 n... | On what theorem is the formula that frequently generates the number 2 and all other primes precisely twice based on? | {
"text": [],
"answer_start": []
} |
5a89357c3b2508001a72a520 | 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... | What is another way to state the condition that infinitely many rows can exist only if a and q are coprime? | {
"text": [],
"answer_start": []
} |
5a89357c3b2508001a72a521 | 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... | If a and q are coprime, which theorem holds that an arithmetic progression has an infinite number of wraps? | {
"text": [],
"answer_start": []
} |
5a89357c3b2508001a72a522 | 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... | What is the density of all wraps compatible with a modulo 9? | {
"text": [],
"answer_start": []
} |
5a89357c3b2508001a72a523 | 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... | If q=9 and a=3, 6 or 9, how many wraps would be in the progression? | {
"text": [],
"answer_start": []
} |
5a89357c3b2508001a72a524 | 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... | If q=9 and a=1,2,4,5, 7, or 8, how many wraps would be in a progression? | {
"text": [],
"answer_start": []
} |
5a8937553b2508001a72a52a | 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., ... | What function is related to Basel numbers? | {
"text": [],
"answer_start": []
} |
5a8937553b2508001a72a52b | 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., ... | What type of value would the Basel function have if there were finite primes? | {
"text": [],
"answer_start": []
} |
5a8937553b2508001a72a52c | 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., ... | What property of the harmonic series 1 + 1/2 + 1/3 +1/4 ... shows that there is an infinite number of Basels? | {
"text": [],
"answer_start": []
} |
5a8937553b2508001a72a52d | 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., ... | What does it mean when a Basel series diverges? | {
"text": [],
"answer_start": []
} |
5a8937553b2508001a72a52e | 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., ... | Of what mathematical nature is the prime problem? | {
"text": [],
"answer_start": []
} |
5a89392f3b2508001a72a534 | 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 r... | When was the function hypothesis proposed? | {
"text": [],
"answer_start": []
} |
5a89392f3b2508001a72a535 | 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 r... | According to the function hypothesis, all zeroes of the ζ-function have real part equal to 1/2 except for what values of s? | {
"text": [],
"answer_start": []
} |
5a89392f3b2508001a72a536 | 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 r... | What does the Riemann hypothesis state the source of irregularity in the distribution of math zeroes from? | {
"text": [],
"answer_start": []
} |
5a89392f3b2508001a72a537 | 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 r... | What type of zero distribution does the Riemann hypothesis propose is also true for short intervals near X? | {
"text": [],
"answer_start": []
} |
5a89392f3b2508001a72a538 | 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 r... | What type of zero distribution is characterized about x/log x of numbers less than x? | {
"text": [],
"answer_start": []
} |
5a893a003b2508001a72a53e | 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 ... | When did Vinogradov propose his four conjectural problems? | {
"text": [],
"answer_start": []
} |
5a893a003b2508001a72a53f | 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 ... | Which conjecture holds that every odd integer n greater than 2 can be expressed as a sum of two primes? | {
"text": [],
"answer_start": []
} |
5a893a003b2508001a72a540 | 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 ... | As of February 2017, how many numbers has Goldbach's conjecture been proven to? | {
"text": [],
"answer_start": []
} |
5a893a003b2508001a72a541 | 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 ... | Which theorem states that all large even integers can be expressed as a sum of three primes? | {
"text": [],
"answer_start": []
} |
5a893a003b2508001a72a542 | 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 ... | Which theorem states that every large odd integer can be written as a prime summed with a semiprime? | {
"text": [],
"answer_start": []
} |
5a893bd73b2508001a72a548 | 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 infinit... | What conjecture holds that there is an infinite amount of twin positives? | {
"text": [],
"answer_start": []
} |
5a893bd73b2508001a72a549 | 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 infinit... | What is a twin positive? | {
"text": [],
"answer_start": []
} |
5a893bd73b2508001a72a54a | 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 infinit... | Which conjecture holds that for any negative integer n, there is an infinite amount of pairs of consecutive primes differing by 2n? | {
"text": [],
"answer_start": []
} |
5a893bd73b2508001a72a54b | 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 infinit... | Of what form is the infinite amount of positives that comprise the special cases of Schinzel's hypothesis? | {
"text": [],
"answer_start": []
} |
5a893bd73b2508001a72a54c | 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 infinit... | What conjecture holds that there are always a minimum of 1 prime between the squares of consecutive primes greater than 2? | {
"text": [],
"answer_start": []
} |
5a893ce63b2508001a72a552 | 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 ... | Besides the study of prime numbers, what general theory was considered the official example of the military? | {
"text": [],
"answer_start": []
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.