doc_id int32 0 2.25M | text stringlengths 101 8.13k | source stringlengths 38 44 |
|---|---|---|
8,600 | On 7 September 2022 during the joint UK/US SinkEx 'Atlantic Thunder' a 41 Squadron Typhoon successfully hit the ex-USS Boone with Paveway IVs, becoming the first RAF Typhoon to strike a naval target with live ordnance. | https://en.wikipedia.org/wiki?curid=167667 |
8,601 | During the 2008 Farnborough Airshow it was announced that Oman was in an "advanced stage" of discussions to order Typhoons as a replacement for its SEPECAT Jaguar aircraft. On 21 December 2012, the Royal Air Force of Oman (RAFO) became the Typhoon's seventh customer when BAE and Oman announced an order for 12 Typhoons to enter service in 2017. The first of the Typhoons (plus Hawk Mk 166) ordered by Oman were "formally presented to the customer" on 15 May 2017. This included a flypast by a RAFO Typhoon. | https://en.wikipedia.org/wiki?curid=167667 |
8,602 | In August 2006, Saudi Arabia confirmed it had agreed to purchase 72 Typhoons for The Royal Saudi Air Force (RSAF). In December 2006, it was reported in "The Guardian" that Saudi Arabia had threatened to buy Rafales because of a UK Serious Fraud Office (SFO) investigation into the Al Yamamah defence deals which commenced in the 1980s. | https://en.wikipedia.org/wiki?curid=167667 |
8,603 | On 14 December 2006, Britain's attorney general, Lord Goldsmith, ordered that the SFO discontinue its investigation into BAE Systems' alleged bribery of senior Saudi officials in the Al-Yamamah contracts, citing "the need to safeguard national and international security". "The Times" raised the possibility that RAF production aircraft would be diverted as early Saudi Arabian aircraft, with the RAF forced to wait for its full complement of aircraft. This arrangement would mirror the diversion of RAF Tornados to the RSAF. "The Times" also reported that such an arrangement would make the UK purchase of its Tranche3 commitments more likely. On 17 September 2007, Saudi Arabia confirmed it had signed a GB£4.43 billion contract for 72 aircraft. 24 aircraft would be at the Tranche2 build standard, previously destined for the UK RAF, the first being delivered in 2008. The remaining 48 aircraft were to be assembled in Saudi Arabia and delivered from 2011, however following contract renegotiations in 2011, it was agreed that all 72 aircraft would be assembled by BAE Systems in the UK, with the last 24 aircraft being built to Tranche3 capability. | https://en.wikipedia.org/wiki?curid=167667 |
8,604 | On 29 September 2008, the United States Department of State approved the Typhoon sale, required because of a certain technology governed by the ITAR process which was incorporated into the MIDS of the Eurofighter. | https://en.wikipedia.org/wiki?curid=167667 |
8,605 | On 22 October 2008, the first RSAF Typhoon made its maiden flight at Warton. Since 2010, BAE has been training Saudi Arabian personnel at Warton. | https://en.wikipedia.org/wiki?curid=167667 |
8,606 | By 2011, 24 Tranche 2 Eurofighter Typhoons had been delivered to Saudi Arabia, consisting of 18 single-seat and six two-seat aircraft. After that, BAE and Riyadh entered into discussions over configurations and price of the rest of the 72-plane order. On 19 February 2014, BAE announced that the Saudis had agreed to a price increase. BAE announced that the last of the original 72 Typhoons had been delivered to Saudi Arabia in June 2017. | https://en.wikipedia.org/wiki?curid=167667 |
8,607 | RSAF Typhoons are playing a central role in the Saudi-led bombing campaign in Yemen. In February 2015, Saudi Typhoons attacked ISIS targets over Syria using Paveway IV bombs for the first time. | https://en.wikipedia.org/wiki?curid=167667 |
8,608 | On 9March 2018, a memorandum of intent for the additional 48 Typhoons was signed during Saudi Crown Prince Mohammed bin Salman's visit to the United Kingdom, however the deal has not been completed due to German arms sanctions implemented in November 2018 in response to the assassination of Jamal Khashoggi. | https://en.wikipedia.org/wiki?curid=167667 |
8,609 | The first Spanish production Eurofighter Tifón to fly was "CE.16-01" (ST001) on 17 February 2003, flying from Getafe Air Base. The Spanish Air and Space Force assigned their Typhoons to QRA responsibilities in July 2008. | https://en.wikipedia.org/wiki?curid=167667 |
8,610 | A Spanish Air and Space Force Typhoon, on a training exercise near Otepää in Estonia, released an AMRAAM missile by mistake on 7August 2018. There were no casualties, but the ten-day search operation for missile remains was unsuccessful and the unknown status of the missile, whether it self-destructed in the air or landed unexploded, left a hazardous situation for the public. The pilot was disciplined for negligence, but received only the minimum penalty in the light of undisclosed mitigating circumstances. | https://en.wikipedia.org/wiki?curid=167667 |
8,611 | The Eurofighter Typhoon was one of the contenders to replace Belgium's fleet of ageing F-16A/B MLU's by 2023. Other contenders include the SAAB Gripen-E/F, Dassault Rafale, F/A-18E/F Super Hornet and F-35A Lightning II. | https://en.wikipedia.org/wiki?curid=167667 |
8,612 | On 25 October 2018, Belgium officially selected the offer for 34 F-35As to replace the current fleet of around 54 F-16s. Government officials said the decision to select the F-35 over the Eurofighter Typhoon came down to price, stating that "The offer from the Americans was the best in all our seven valuation criteria." The total purchasing price for the aircraft and its support until 2030 totaled €4billion, €600million cheaper than the initially budgeted €4.6billion. | https://en.wikipedia.org/wiki?curid=167667 |
8,613 | The Royal Danish Air Force held a competition to replace its ageing fleet of F-16s in which the Eurofighter Typhoon, Boeing F/A-18F Super Hornet and the F-35 Lightning II were assessed. Denmark is a level-3 partner in the Joint Strike Fighter programme, and had already invested $200million. On 12 May 2016 the Danish government recommended that 27 F-35A fighters, instead of 34 Typhoons, should be procured. | https://en.wikipedia.org/wiki?curid=167667 |
8,614 | In 2005 the Eurofighter was a contender for Singapore's next generation fighter requirement competing with the Boeing F-15SG and the Dassault Rafale. The Eurofighter was eliminated from the competition in June 2005. | https://en.wikipedia.org/wiki?curid=167667 |
8,615 | In 2002, the Republic of Korea Air Force (ROKAF) chose the F-15K Slam Eagle over the Dassault Rafale, Eurofighter Typhoon and Sukhoi Su-35 for its 40 aircraft F-X Phase I fighter competition. During 2012–13, the Typhoon competed with the Boeing F-15SE Silent Eagle and the F-35 for the ROKAF's F-X Phase III fighter competition. In November 2013, it was announced that the ROKAF will purchase 40 F-35As. | https://en.wikipedia.org/wiki?curid=167667 |
8,616 | According to "Eurofighter World" magazine, Bangladesh, Egypt, Finland and Switzerland were among countries interested in acquiring the Eurofighter Typhoon. In mid-2021, it was reported that the Lockheed Martin F-35 was selected in Switzerland's $6.5 billion fighter competition, beating bids from Eurofighter, Dassault, and Boeing. | https://en.wikipedia.org/wiki?curid=167667 |
8,617 | Finland was offered partner status in the Eurofighter Typhoon programme as part of the consortium's bid for Finland's HX Fighter Program competition, reported British magazine "Janes" on 6 July 2021. As part of Finland's HX offering, BAE Systems proposed a new Large Area Display (LAD) to replace the three multi-function head-down displays (MHDDs). On 10 December 2021 Finland officially selected the F-35A as the country's next fighter. | https://en.wikipedia.org/wiki?curid=167667 |
8,618 | The Eurofighter is produced in single-seat and twin-seat variants. The twin-seat variant is not used operationally, but only for training, though it is combat capable. The aircraft has been manufactured in three major standards; seven Development Aircraft (DA), seven production standard Instrumented Production Aircraft (IPA) for further system development and a continuing number of Series Production Aircraft. The production aircraft are now operational with the partner nation's air forces. | https://en.wikipedia.org/wiki?curid=167667 |
8,619 | The Tranche 1 aircraft were produced from 2000 onwards. Aircraft capabilities are being increased incrementally, with each software upgrade resulting in a different standard, known as blocks. With the introduction of the block5 standard, the R2 retrofit programme began to bring all Tranche1 aircraft to that standard. | https://en.wikipedia.org/wiki?curid=167667 |
8,620 | A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. | https://en.wikipedia.org/wiki?curid=23666 |
8,621 | However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. | https://en.wikipedia.org/wiki?curid=23666 |
8,622 | The property of being prime is called primality. A simple but slow method of checking the primality of a given number formula_1, called trial division, tests whether formula_1 is a multiple of any integer between 2 and formula_3. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance 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. the largest known prime number is a Mersenne prime with 24,862,048 decimal digits. | https://en.wikipedia.org/wiki?curid=23666 |
8,623 | There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically 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 of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its logarithm. | https://en.wikipedia.org/wiki?curid=23666 |
8,624 | Several historical questions regarding prime numbers are still unsolved. These include 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 having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, objects that behave in a generalized way like prime numbers include prime elements and prime ideals. | https://en.wikipedia.org/wiki?curid=23666 |
8,625 | A natural number (1, 2, 3, 4, 5, 6, etc.) is called a "prime number" (or a "prime") if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite numbers. In other words, formula_1 is prime if formula_1 items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange formula_1 dots into a rectangular grid that is more than one dot wide and more than one dot high. | https://en.wikipedia.org/wiki?curid=23666 |
8,626 | For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). | https://en.wikipedia.org/wiki?curid=23666 |
8,627 | Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisors, 1 and the number itself. | https://en.wikipedia.org/wiki?curid=23666 |
8,628 | Yet another way to express the same thing is that a number formula_1 is prime if it is greater than one and if none of the numbers formula_10 divides formula_1 evenly. | https://en.wikipedia.org/wiki?curid=23666 |
8,629 | No even number formula_1 greater than 2 is prime because any such number can be expressed as the product formula_13. Therefore, every prime number other than 2 is an odd number, and is called an "odd prime". Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: | https://en.wikipedia.org/wiki?curid=23666 |
8,630 | decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5. | https://en.wikipedia.org/wiki?curid=23666 |
8,631 | The set of all primes is sometimes denoted by formula_14 (a boldface capital "P") or by formula_15 (a blackboard bold capital P). | https://en.wikipedia.org/wiki?curid=23666 |
8,632 | The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from ancient Greek mathematics. Euclid's "Elements" (c. 300 BC) proves the infinitude of primes and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime. Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of | https://en.wikipedia.org/wiki?curid=23666 |
8,633 | Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem, characterizing the prime numbers as the numbers formula_1 that evenly divide formula_17. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it. Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit. Fibonacci brought the innovations from Islamic mathematics back to Europe. His book "Liber Abaci" (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root. | https://en.wikipedia.org/wiki?curid=23666 |
8,634 | In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Fermat also investigated the primality of the and Marin Mersenne studied the Mersenne primes, prime numbers of the form formula_18 with formula_19 itself a prime. Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes formula_20. | https://en.wikipedia.org/wiki?curid=23666 |
8,635 | At the start of the 19th century, Legendre and Gauss conjectured that as formula_21 tends to infinity, the number of primes up to formula_21 is asymptotic to formula_23, where formula_24 is the natural logarithm of formula_21. A weaker consequence of this high density of primes was Bertrand's postulate, that for every formula_26 there is a prime between formula_1 and formula_28, proved in 1852 by Pafnuty Chebyshev. Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem. Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,636 | Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877), Proth's theorem (c. 1878), the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test. | https://en.wikipedia.org/wiki?curid=23666 |
8,637 | Since 1951 all the largest known primes have been found using these tests on computers. The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects. The idea that prime numbers had few applications outside of pure mathematics was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis. | https://en.wikipedia.org/wiki?curid=23666 |
8,638 | The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size. | https://en.wikipedia.org/wiki?curid=23666 |
8,639 | Most early Greeks did not even consider 1 to be a number, so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including Nicomachus, Iamblichus, Boethius, and Cassiodorus also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number. | https://en.wikipedia.org/wiki?curid=23666 |
8,640 | By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number. In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956. | https://en.wikipedia.org/wiki?curid=23666 |
8,641 | If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with any number of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1. By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit". | https://en.wikipedia.org/wiki?curid=23666 |
8,642 | Writing a number as a product of prime numbers is called a "prime factorization" of the number. For example: | https://en.wikipedia.org/wiki?curid=23666 |
8,643 | The terms in the product are called "prime factors". The same prime factor may occur more than once; this example has two copies of the prime factor formula_30 When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, formula_31 denotes the square or second power of formula_30 | https://en.wikipedia.org/wiki?curid=23666 |
8,644 | The central importance of prime numbers to number theory and mathematics in general stems from the "fundamental theorem of arithmetic". This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, | https://en.wikipedia.org/wiki?curid=23666 |
8,645 | this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes, | https://en.wikipedia.org/wiki?curid=23666 |
8,646 | although their ordering may differ. So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers. | https://en.wikipedia.org/wiki?curid=23666 |
8,647 | Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If formula_19 is a prime number and formula_19 divides a product formula_35 of integers formula_36 and formula_37 then formula_19 divides formula_36 or formula_19 divides formula_41 (or both). Conversely, if a number formula_19 has the property that when it divides a product it always divides at least one factor of the product, then formula_19 must be prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,648 | of prime numbers never ends. This statement is referred to as "Euclid's theorem" in honor of the ancient Greek mathematician Euclid, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on Fermat numbers, Furstenberg's proof using general topology, and Kummer's elegant proof. | https://en.wikipedia.org/wiki?curid=23666 |
8,649 | Euclid's proof shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add formula_44 If the list consists of the primes formula_45 this gives the number | https://en.wikipedia.org/wiki?curid=23666 |
8,650 | with one or more prime factors. formula_47 is evenly divisible by each of these factors, but formula_47 has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of formula_47 can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,651 | The numbers formed by adding one to the products of the smallest primes are called Euclid numbers. The first five of them are prime, but the sixth, | https://en.wikipedia.org/wiki?curid=23666 |
8,652 | There is no known efficient formula for primes. For example, there is no non-constant polynomial, even in several variables, that takes "only" prime values. However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. There is also a set of Diophantine equations in nine variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its "positive" values are prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,653 | Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants formula_53 and formula_54 such that | https://en.wikipedia.org/wiki?curid=23666 |
8,654 | are prime for any natural number formula_1 in the first formula, and any number of exponents in the second formula. Here formula_57 represents the floor function, the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of formula_58 or formula_59 | https://en.wikipedia.org/wiki?curid=23666 |
8,655 | Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood 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 formula_1 greater than 2 can be written as a sum of two primes. , this conjecture has been verified for all numbers up to formula_61 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 greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. | https://en.wikipedia.org/wiki?curid=23666 |
8,656 | The existence of arbitrarily large prime gaps can be seen by noting that the sequence formula_62 consists of formula_63 composite numbers, for any natural number formula_64 However, large prime gaps occur much earlier than this argument shows. For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than formula_65 It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2; this is the twin prime conjecture. Polignac's conjecture states more generally that for every positive integer formula_66 there are infinitely many pairs of consecutive primes that differ by formula_67 | https://en.wikipedia.org/wiki?curid=23666 |
8,657 | Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture all suggest that the largest gaps between primes from formula_68 to formula_1 should be at most approximately formula_70 a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at formula_71 Prime gaps can be generalized to prime formula_72-tuples, patterns in the differences between more than two prime numbers. Their infinitude and density are the subject of the first Hardy–Littlewood conjecture, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem. | https://en.wikipedia.org/wiki?curid=23666 |
8,658 | Analytic number theory studies number theory through the lens of continuous functions, limits, infinite series, and the related mathematics of the infinite and infinitesimal. | https://en.wikipedia.org/wiki?curid=23666 |
8,659 | This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem. | https://en.wikipedia.org/wiki?curid=23666 |
8,660 | which today can be recognized as the value formula_74 of the Riemann zeta function. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis. Euler showed that formula_75. | https://en.wikipedia.org/wiki?curid=23666 |
8,661 | The reciprocal of this number, formula_76, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common). | https://en.wikipedia.org/wiki?curid=23666 |
8,662 | The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the formula_1-th prime is known. | https://en.wikipedia.org/wiki?curid=23666 |
8,663 | with relatively prime integers formula_36 and formula_41 take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same formula_41 have approximately the same proportions of primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,664 | Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often. | https://en.wikipedia.org/wiki?curid=23666 |
8,665 | Euler showed that, for any arbitrary real number formula_21, there exists a prime formula_19 for which this sum is bigger than formula_21. This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every formula_21. | https://en.wikipedia.org/wiki?curid=23666 |
8,666 | The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum | https://en.wikipedia.org/wiki?curid=23666 |
8,667 | does not grow to infinity as formula_1 goes to infinity (see the Basel problem). In this sense, prime numbers occur more often than squares of natural numbers, | https://en.wikipedia.org/wiki?curid=23666 |
8,668 | although both sets are infinite. Brun's theorem states that the sum of the reciprocals of twin primes, | https://en.wikipedia.org/wiki?curid=23666 |
8,669 | is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the twin prime conjecture, that there exist infinitely many twin primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,670 | The prime-counting function formula_90 is defined as the number of primes not greater than formula_1. For example, formula_92, since there are five primes less than or equal to 11. Methods such as the Meissel–Lehmer algorithm can compute exact values of formula_90 faster than it would be possible to list each prime up to formula_1. The prime number theorem states that formula_90 is asymptotic to formula_96, which is denoted as | https://en.wikipedia.org/wiki?curid=23666 |
8,671 | and means that the ratio of formula_90 to the right-hand fraction approaches 1 as formula_1 grows to infinity. This implies that the likelihood that a randomly chosen number less than formula_1 is prime is (approximately) inversely proportional to the number of digits in formula_1. | https://en.wikipedia.org/wiki?curid=23666 |
8,672 | An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. This difference is called the modulus of the progression. For example, | https://en.wikipedia.org/wiki?curid=23666 |
8,673 | is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression | https://en.wikipedia.org/wiki?curid=23666 |
8,674 | can have more than one prime only when its remainder formula_36 and modulus formula_109 are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,675 | The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes. | https://en.wikipedia.org/wiki?curid=23666 |
8,676 | yields prime numbers for formula_111, although composite numbers appear among its later values. The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients | https://en.wikipedia.org/wiki?curid=23666 |
8,677 | in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values. | https://en.wikipedia.org/wiki?curid=23666 |
8,678 | The Ulam spiral arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others. | https://en.wikipedia.org/wiki?curid=23666 |
8,679 | One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the Riemann zeta function formula_112 are located. | https://en.wikipedia.org/wiki?curid=23666 |
8,680 | This function is an analytic function on the complex numbers. For complex numbers formula_113 with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers, | https://en.wikipedia.org/wiki?curid=23666 |
8,681 | This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers. | https://en.wikipedia.org/wiki?curid=23666 |
8,682 | then the sum-product equality would also be valid at formula_115, but the sum would diverge (it is the harmonic series formula_116) while the product would be finite, a contradiction. | https://en.wikipedia.org/wiki?curid=23666 |
8,683 | The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the prime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1, although other more elementary proofs have been found. | https://en.wikipedia.org/wiki?curid=23666 |
8,684 | The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. | https://en.wikipedia.org/wiki?curid=23666 |
8,685 | In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the | https://en.wikipedia.org/wiki?curid=23666 |
8,686 | asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of formula_21 for intervals near a number formula_21). | https://en.wikipedia.org/wiki?curid=23666 |
8,687 | Modular arithmetic modifies usual arithmetic by only using the numbers formula_119, for a natural number formula_1 called the modulus. | https://en.wikipedia.org/wiki?curid=23666 |
8,688 | Any other natural number can be mapped into this system by replacing it by its remainder after division by formula_1. | https://en.wikipedia.org/wiki?curid=23666 |
8,689 | Modular sums, differences and products are calculated by performing the same replacement by the remainder | https://en.wikipedia.org/wiki?curid=23666 |
8,690 | on the result of the usual sum, difference, or product of integers. Equality of integers corresponds to "congruence" in modular arithmetic: | https://en.wikipedia.org/wiki?curid=23666 |
8,691 | formula_21 and formula_123 are congruent (written formula_124 mod formula_1) when they have the same remainder after division by formula_1. However, in this system of numbers, division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number formula_127 as modulus, division by formula_128 is possible: formula_129, because clearing denominators by multiplying both sides by formula_128 gives the valid formula formula_131. However, with the composite modulus formula_132, division by formula_128 is impossible. There is no valid solution to formula_134: clearing denominators by multiplying by formula_128 causes the left-hand side to become formula_136 while the right-hand side becomes either formula_137 or formula_128. | https://en.wikipedia.org/wiki?curid=23666 |
8,692 | In the terminology of abstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a ring but not a field. | https://en.wikipedia.org/wiki?curid=23666 |
8,693 | Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if | https://en.wikipedia.org/wiki?curid=23666 |
8,694 | Giuga's conjecture says that this equation is also a sufficient condition for formula_19 to be prime. | https://en.wikipedia.org/wiki?curid=23666 |
8,695 | Wilson's theorem says that an integer formula_147 is prime if and only if the factorial formula_148 is congruent to formula_149 mod formula_19. For a composite this cannot hold, since one of its factors divides both and formula_151, and so formula_152 is impossible. | https://en.wikipedia.org/wiki?curid=23666 |
8,696 | The formula_19-adic order formula_154 of an integer formula_1 is the number of copies of formula_19 in the prime factorization of formula_1. The same concept can be extended from integers to rational numbers by defining the formula_19-adic order of a fraction formula_159 to be formula_160. The formula_19-adic absolute value formula_162 of any rational number formula_109 is then defined as | https://en.wikipedia.org/wiki?curid=23666 |
8,697 | formula_164. Multiplying an integer by its formula_19-adic absolute value cancels out the factors of formula_19 in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their formula_19-adic distance, the formula_19-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of formula_19. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a complete field, the rational numbers with the formula_19-adic distance can be extended to a different complete field, the formula_19-adic numbers. | https://en.wikipedia.org/wiki?curid=23666 |
8,698 | This picture of an order, absolute value, and complete field derived from them can be generalized to algebraic number fields and their valuations (certain mappings from the multiplicative group of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings from the field to the real numbers, also called norms), and places (extensions to complete fields in which the given field is a dense set, also called completions). The extension from the rational numbers to the real numbers, for instance, is a place in which the distance between numbers is the usual absolute value of their difference. The corresponding mapping to an additive group would be the logarithm of the absolute value, although this does not meet all the requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and formula_19-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers. The local-global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory. | https://en.wikipedia.org/wiki?curid=23666 |
8,699 | A commutative ring is an algebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, "prime elements" and "irreducible elements". An element formula_19 of a ring formula_174 is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a unit), and satisfies the following requirement: whenever formula_19 divides the product formula_176 of two elements of formula_174, it also divides at least one of formula_21 or formula_123. An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set, | https://en.wikipedia.org/wiki?curid=23666 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.