Contents 1 Definition and examples 2 History 2.1 Primality of one 3 Elementary properties 3.1 Unique factorization 3.2 Infiniteness 3.3 Formulas for primes 3.4 Open questions 4 Analytic properties 4.1 Analytical proof of Euclid's theorem 4.2 Number of prime numbers below a given number 4.3 Arithmetic progressions 4.4 Prime values of quadratic polynomials 4.5 Zeta function and the Riemann hypothesis 5 Abstract algebra 5.1 Modular arithmetic and finite fields 5.2 p-adic numbers 5.3 Prime elements in rings 5.4 Prime ideals 5.5 Group theory 6 Computational methods 6.1 Trial division 6.2 Sieves 6.3 Primality testing versus primality proving 6.4 Special-purpose algorithms and the largest known prime 6.5 Integer factorization 6.6 Other computational applications 7 Related topics 7.1 Constructible polygons and polygon partitions 7.2 Quantum mechanics 7.3 Biology 7.4 Arts and literature 8 Notes 9 References 10 External links 10.1 Generators and calculators

Definition and examples Main article: List of prime numbers 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 a product of two natural numbers that are both smaller than it. The numbers greater than 1 that are not prime are called composite numbers.[1] In other words, n {\displaystyle n} is prime if n {\displaystyle n} items cannot be divided up into smaller equal-size groups of more than one item,[2] or if it is not possible to arrange n {\displaystyle n} dots into a rectangular grid that is more than one dot wide and more than one dot high.[3] For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,[4] as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. 4 = 2 × 2 and 6 = 2 × 3 are both composite. Demonstration, with Cuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly The divisors of a natural number n {\displaystyle n} are the numbers that divide n {\displaystyle n} evenly. 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.[5] Yet another way to say the same thing is that a number n {\displaystyle n} is prime if it is greater than one and if none of the numbers 2 , 3 , … , n − 1 {\displaystyle 2,3,\dots ,n-1} divides n {\displaystyle n} evenly.[6] The first 25 prime numbers (all the prime numbers less than 100) are:[7] 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (sequence A000040 in the OEIS). No even number n {\displaystyle n} greater than 2 is prime because any such number can be expressed as the product 2 × n / 2 {\displaystyle 2\times n/2} . Therefore, every prime number other than 2 is an odd number, and is called an odd prime.[8] Similarly, when written in the usual decimal system, all prime numbers larger than 5 must end in 1, 3, 7, or 9. The numbers that end with other digits are all composite: 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.[9] The set of all primes is sometimes denoted by P {\displaystyle \mathbf {P} } (a boldface capital P)[10] or by P {\displaystyle \mathbb {P} } (a blackboard bold capital P).[11]

History The Ishango bone, from the Upper Paleolithic era, is suggested to include a list of primes The Rhind Mathematical Papyrus It has been suggested that engravings on the Ishango bone, from about 19,000 BC, record a list of prime numbers.[12] The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers.[13] However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. Euclid's Elements (circa 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.[14] Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes.[15][16] Around 1000 AD, the Islamic mathematician Alhazen found Wilson's theorem, characterizing the prime numbers as the numbers n {\displaystyle n} that evenly divide ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1} . Alhazen also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.[17] Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by testing only the divisors up to the square root of the largest number to be tested. 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.[16] In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler).[18] Fermat also investigated the primality of the Fermat numbers 2 2 n + 1 {\displaystyle 2^{2^{n}}+1} ,[19] and Marin Mersenne studied the Mersenne primes, prime numbers of the form 2 p − 1 {\displaystyle 2^{p}-1} with p {\displaystyle p} itself a prime.[20] Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.[21] Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes.[14] 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 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + … {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\dots } .[22] At the start of the 19th century, Legendre and Gauss conjectured that as x {\displaystyle x} tends to infinity, the number of primes up to x {\displaystyle x} is asymptotic to x / ln ⁡ x {\displaystyle x/\ln x} , where ln ⁡ x {\displaystyle \ln x} is the natural logarithm of x {\displaystyle x} . Ideas of Riemann in his 1859 paper on the zeta-function sketched an outline for proving this. 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.[23] Another important 19th-century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.[24] 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),[25] Proth's theorem (around 1878),[26] the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.[16] Since 1951 all the largest known primes have been found using these tests on computers.[a] The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects.[7][28] The idea that prime numbers had few applications outside of pure mathematics[b] was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.[31] 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.[15][32][33] The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) on long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.[34] Primality of one Most early Greeks did not even consider 1 to be a number,[35][36] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. 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.[35] By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number.[37] 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.[38] In the 19th century many mathematicians still considered 1 to be prime,[39] and lists of primes that included 1 continued to be published as recently as 1956.[40][41] 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 different numbers of copies of 1.[39] 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.[41] 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.[42] 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".[39]

Elementary properties Unique factorization Main article: Fundamental theorem of arithmetic Writing a number as a product of prime numbers is called a prime factorization of the number. For example: 34866 = 2 ⋅ 3 ⋅ 3 ⋅ 13 ⋅ 149 = 2 ⋅ 3 2 ⋅ 13 ⋅ 149 {\displaystyle {\begin{aligned}34866&=2\cdot 3\cdot 3\cdot 13\cdot 149\\&=2\cdot 3^{2}\cdot 13\cdot 149\end{aligned}}} 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 3 {\displaystyle 3} . When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for instance, in the second way of writing the product above, 3 2 {\displaystyle 3^{2}} denotes the square or second power of 3 {\displaystyle 3} . The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.[43] This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, 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, although their ordering may differ.[44] 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.[45] Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If p {\displaystyle p} is a prime number and p {\displaystyle p} divides a product a b {\displaystyle ab} of integers a {\displaystyle a} and b {\displaystyle b} , then p {\displaystyle p} divides a {\displaystyle a} or p {\displaystyle p} divides b {\displaystyle b} (or both).[46] Conversely, if a number p {\displaystyle p} has the property that when it divides a product it always divides at least one factor of the product, then p {\displaystyle p} must be prime.[47] Infiniteness Main article: Euclid's theorem There are infinitely many prime numbers. Another way of saying this is that the sequence 2, 3, 5, 7, 11, 13, ... 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,[48] Furstenberg's proof using general topology,[49] and Kummer's elegant proof.[50] Euclid's proof[51] shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add 1 {\displaystyle 1} . If the list consists of the primes p 1 , p 2 , … p n {\displaystyle p_{1},p_{2},\dots p_{n}} , this gives the number N = 1 + p 1 ⋅ p 2 ⋯ p n . {\displaystyle N=1+p_{1}\cdot p_{2}\cdots p_{n}.} By the fundamental theorem, N {\displaystyle N} has a prime factorization N = p 1 ′ ⋅ p 2 ′ ⋯ p m ′ {\displaystyle N=p'_{1}\cdot p'_{2}\cdots p'_{m}} with one or more prime factors. N {\displaystyle N} is evenly divisible by each of these factors, but N {\displaystyle N} has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of N {\displaystyle N} can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes. The numbers formed by adding one to the products of the smallest primes are called Euclid numbers.[52] The first five of them are prime, but the sixth, 30031 = 1 + 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 13 = 59 ⋅ 509 , {\displaystyle 30031=1+2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13=59\cdot 509,} is a composite number. Formulas for primes Main article: Formula for primes 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.[53] 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.[54] 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.[53] Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants A > 1 {\displaystyle A>1} and μ {\displaystyle \mu } such that ⌊ A 3 n ⌋  and  ⌊ 2 ⋯ 2 2 μ ⌋ {\displaystyle \left\lfloor A^{3^{n}}\right\rfloor {\text{ and }}\left\lfloor 2^{\cdots ^{2^{2^{\mu }}}}\right\rfloor } are prime for any natural number n {\displaystyle n} in the first formula, and any number of exponents in the second formula.[55] Here ⌊ ⋅ ⌋ {\displaystyle \lfloor \,\cdot \,\rfloor } 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 A {\displaystyle A} or μ {\displaystyle \mu } .[53] Open questions Further information: Category:Conjectures about prime numbers 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.[56] One of them is Goldbach's conjecture, which asserts that every even integer n {\displaystyle n} greater than 2 can be written as a sum of two primes.[57] As of 2014[update], this conjecture has been verified for all numbers up to n = 4 ⋅ 10 18 {\displaystyle n=4\cdot 10^{18}} .[58] 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.[59] 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.[60] Also, any even integer can be written as the sum of six primes.[61] The branch of number theory studying such questions is called additive number theory.[62] Another type of problem concerns prime gaps, the differences between consecutive primes. The existence of arbitrarily large prime gaps can be seen by noting that the sequence n ! + 2 , n ! + 3 , … , n ! + n {\displaystyle n!+2,n!+3,\dots ,n!+n} consists of n − 1 {\displaystyle n-1} composite numbers, for any natural number n {\displaystyle n} .[63] However, large prime gaps occur much earlier than this argument shows.[64] For example, the first prime gap of length 8 is between the primes 89 and 97,[65] much smaller than 8 ! = 40320 {\displaystyle 8!=40320} . 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 k {\displaystyle k} , there are infinitely many pairs of consecutive primes that differ by 2 k {\displaystyle 2k} .[66] Andrica's conjecture,[66] Brocard's conjecture,[67] Legendre's conjecture,[68] and Oppermann's conjecture[67] all suggest that the largest gaps between primes from 1 {\displaystyle 1} to n {\displaystyle n} should be at most approximately n {\displaystyle {\sqrt {n}}} , a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér's conjecture sets the largest gap size at O ( log 2 ⁡ n ) {\displaystyle O(\log ^{2}n)} .[66] Prime gaps can be generalized to prime k {\displaystyle k} -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.[69]

Computational methods The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13. 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 mathematics[b] with the exception of use of prime numbered gear teeth to distribute wear evenly.[116] In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.[117] This vision of the purity of number theory 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.[31] These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining with a given number is prime. The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types. Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments (or all of its prime factors) are called factorization algorithms. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators. Trial division Main article: Trial division The most basic method of checking the primality of a given integer n {\displaystyle n} is called trial division. This method divides n {\displaystyle n} by each integer from 2 up to the square root of n {\displaystyle n} . Any such integer dividing n {\displaystyle n} evenly establishes n {\displaystyle n} as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever n = a ⋅ b {\displaystyle n=a\cdot b} , one of the two factors a {\displaystyle a} and b {\displaystyle b} is less than or equal to the square root of n {\displaystyle n} . Another optimization is to check only primes as factors in this range.[118] For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to √37, which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers.[119] However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter.[120] Sieves The sieve of Eratosthenes starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc) are marked as primes (magenta). Main article: Sieve of Eratosthenes Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed.[121] The oldest method for generating a list of primes is called the sieve of Eratosthenes.[122] The animation shows an optimized variant of this method.[123] Another more efficient sieving method for the same problem is the sieve of Atkin.[124] In advanced mathematics, sieve theory applies similar methods to other problems.[125] Primality testing versus primality proving Some of the fastest modern tests for whether an arbitrary given number n {\displaystyle n} is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer.[126] For instance the Solovay–Strassen primality test on a given number p {\displaystyle p} chooses a number a {\displaystyle a} randomly from 1 {\displaystyle 1} to p − 1 {\displaystyle p-1} and uses modular exponentiation to check whether a ( p − 1 ) / 2 ± 1 {\displaystyle a^{(p-1)/2}\pm 1} is divisible by p {\displaystyle p} .[c] If so, it answers yes and otherwise it answers no. If p {\displaystyle p} really is prime, it will always answer yes, but if p {\displaystyle p} is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2.[127] If this test is repeated n {\displaystyle n} times on the same number, the probability that a composite number could pass the test every time is at most 1 / 2 n {\displaystyle 1/2^{n}} . Because this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite.[128] A composite number that passes such a test is called a pseudoprime.[127] In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms, such as the AKS primality test,[129] and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving.[126] The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice.[130] These methods can be used to generate large random prime numbers, by generating and testing random numbers until finding one that is prime; when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime.[d] The following table lists some of these tests. Their running time is given in terms of n {\displaystyle n} , the number to be tested and, for probabilistic algorithms, the number k {\displaystyle k} of tests performed. Moreover, ε {\displaystyle \varepsilon } is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters n {\displaystyle n} and k {\displaystyle k} . Test Developed in Type Running time Notes References AKS primality test 2002 deterministic O ( ( log ⁡ n ) 6 + ε ) {\displaystyle O((\log n)^{6+\varepsilon })} [129][132] Elliptic curve primality proving 1977 Las Vegas O ( ( log ⁡ n ) 4 + ε ) {\displaystyle O((\log n)^{4+\varepsilon })} heuristically [130] Miller–Rabin primality test 1980 Monte Carlo O ( k ( log ⁡ n ) 2 + ε ) {\displaystyle O(k(\log n)^{2+\varepsilon })} error probability 4 − k {\displaystyle 4^{-k}} [133] Solovay–Strassen primality test 1977 Monte Carlo O ( k ( log ⁡ n ) 2 + ε ) {\displaystyle O(k(\log n)^{2+\varepsilon })} error probability 2 − k {\displaystyle 2^{-k}} [133] Special-purpose algorithms and the largest known prime Further information: List of prime numbers In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the Lucas–Lehmer primality test can determine whether a Mersenne number (one less than a power of two) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test.[134] This is why since 1992 (as of January 2018[update]) the largest known prime has always been a Mersenne prime.[135] It is conjectured that there are infinitely many Mersenne primes.[136] The following table gives the largest known primes of various 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.[137] The Electronic Frontier Foundation also offers$150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.[138] Type Prime Number of decimal digits Date Found by Mersenne prime 277,232,917 − 1 23,249,425 December 26, 2017[139] Jonathan Pace, Great Internet Mersenne Prime Search not a Mersenne prime (Proth number) 10,223 × 231,172,165 + 1 9,383,761 October 31, 2016[140] Péter Szabolcs, PrimeGrid[141] factorial prime 208,003! − 1 1,015,843 July 2016 Sou Fukui[142] primorial prime[e] 1,098,133# − 1 476,311 March 2012 James P. Burt, PrimeGrid[144] twin primes 2,996,863,034,895 × 21,290,000 ± 1 388,342 September 2016 Tom Greer, PrimeGrid[145] Integer factorization Main article: Integer factorization Given a composite integer n {\displaystyle n} , the task of providing one (or all) prime factors is referred to as factorization of n {\displaystyle n} . It is significantly more difficult than primality testing,[146] and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and Pollard's rho algorithm can be used to find very small factors of n {\displaystyle n} ,[147] and elliptic curve factorization can be effective when n {\displaystyle n} has factors of moderate size.[148] Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve.[149] As of January 2018[update], the largest number known to have been factored by a general-purpose algorithm is RSA-768, which has 232 decimal digits (768 bits) and is the product of two large primes.[150] Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer.[151] However current technology can only run this algorithm for very small numbers. The largest number that has been factored by a quantum computer running this algorithm is 21.[152] Other computational applications Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (2048-bit primes are common).[153] RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers x {\displaystyle x} and y {\displaystyle y} than to calculate x {\displaystyle x} and y {\displaystyle y} (assumed coprime) if only the product x y {\displaystyle xy} is known.[31] The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing a b mod c {\displaystyle a^{b}{\bmod {c}}} ), while the reverse operation (the discrete logarithm) is thought to be a hard problem.[154] Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to k {\displaystyle k} -independent hashing by using higher-degree polynomials, again modulo large primes.[155] As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table.[156] Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits.[157] Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than 2 16 {\displaystyle 2^{16}} .[158] Prime numbers are also used in pseudorandom number generators including linear congruential generators[159] and the Mersenne Twister.[160] Related topics Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area.[161] Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square.[162] The connected sum of two prime knots 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 of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name.[163] 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 connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.[164] The prime decomposition of 3-manifolds is another example of this type.[165] Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas. Constructible polygons and polygon partitions Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a Fermat prime. Fermat primes are primes of the form F k = 2 2 k + 1 , {\displaystyle F_{k}=2^{2^{k}}+1,} with k {\displaystyle k} a natural number.[166] They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime,[167] but F 5 {\displaystyle F_{5}} is composite and so are all other Fermat numbers that have been verified as of 2017.[168] A regular n {\displaystyle n} -gon is constructible using straightedge and compass if and only if the odd prime factors of n {\displaystyle n} (if any) are distinct Fermat primes.[167] Likewise, a regular n {\displaystyle n} -gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of n {\displaystyle n} are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form 2 a 3 b + 1 {\displaystyle 2^{a}3^{b}+1} .[169] It is possible to partition any convex polygon into n {\displaystyle n} smaller convex polygons of equal area and equal perimeter, when n {\displaystyle n} is a power of a prime number, but this is not known for other values of n {\displaystyle n} .[170] Quantum mechanics Beginning with the work of Hugh Montgomery and Freeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems.[171][172] Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures.[173][174] Biology The evolutionary strategy used by cicadas of the genus Magicicada makes use of prime numbers.[175] 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. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators to synchronize with these cycles.[176][177] In contrast, the multi-year periods between flowering in bamboo plants are hypothesized to be smooth numbers, having only small prime numbers in their factorizations.[178] Arts and literature 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".[179] In his science fiction novel Contact, scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975.[180] In the novel The Curious Incident of the Dog in the Night-Time by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome.[181] Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.[182] Notes ^ A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.[27] ^ a b For instance, Beiler writes that number theorist Ernst Kummer loved his ideal numbers, closely related to the primes, "because they had not soiled themselves with any practical applications",[29] and Katz writes that Edmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as geometry that had already shown themselves to be useful.[30] ^ In this test, the ± 1 {\displaystyle \pm 1} term is negative if a {\displaystyle a} is a square modulo the given (supposed) prime p {\displaystyle p} , and positive otherwise. More generally, for non-prime values of p {\displaystyle p} , the ± 1 {\displaystyle \pm 1} term is the (negated) Jacobi symbol, which can be calculated using quadratic reciprocity. ^ Indeed, much of the analysis of elliptic curve primality proving is based on the assumption that the input to the algorithm has already passed a probabilistic test.[131] ^ The primorial function of n {\displaystyle n} , denoted by n # {\displaystyle n\#} , yields the product of the prime numbers up to n {\displaystyle n} , and a primorial prime is a prime of one of the forms n # ± 1 {\displaystyle n\#\pm 1} .[143] References ^ Gardiner, Anthony (1997). The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965–1996. Oxford University Press. p. 26. ISBN 9780198501053. ^ Henderson, Anne (2014). Dyslexia, Dyscalculia and Mathematics: A practical guide (2nd ed.). Routledge. p. 62. ISBN 9781136636622. ^ Adler, Irving (1960). The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space. Golden Press. p. 16. OCLC 6975809. ^ Leff, Lawrence S. (2000). Math Workbook for the SAT I. Barron's Educational Series. p. 360. ISBN 9780764107689. ^ Dudley, Underwood (1978). "Section 2: Unique factorization". Elementary number theory (2nd ed.). W. H. Freeman and Co. p. 10. ISBN 978-0-7167-0076-0. ^ Sierpiński, Wacław (1988). Elementary Theory of Numbers. North-Holland Mathematical Library. 31 (2nd ed.). Elsevier. p. 113. ISBN 9780080960197. ^ a b Ziegler, Günter M. (2004). "The great prime number record races". Notices of the American Mathematical Society. 51 (4): 414–416. MR 2039814. ^ Stillwell, John (1997). Numbers and Geometry. Undergraduate Texts in Mathematics. Springer. p. 9. ISBN 9780387982892. ^ Sierpiński, Wacław (1964). A Selection of Problems in the Theory of Numbers. New York: Macmillan. p. 40. MR 0170843. ^ Nathanson, Melvyn B. (2000). "Notations and Conventions". Elementary Methods in Number Theory. Graduate Texts in Mathematics. 195. Springer. ISBN 9780387227382. MR 1732941. ^ Faticoni, Theodore G. (2012). The Mathematics of Infinity: A Guide to Great Ideas. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. 111 (2nd ed.). John Wiley & Sons. p. 44. ISBN 9781118243824. ^ Everett, Caleb (2017). Numbers and the Making of Us: Counting and the Course of Human Cultures. Harvard University Press. p. 35. ISBN 9780674504431. ^ Bruins, Evert Marie, review in Mathematical Reviews of Gillings, R. J. (1974). "The recto of the Rhind Mathematical Papyrus. How did the ancient Egyptian scribe prepare it?". Archive for History of Exact Sciences. 12: 291–298. doi:10.1007/BF01307175. MR 0497458. ^ a b Stillwell, John (2010). Mathematics and Its History. Undergraduate Texts in Mathematics (3rd ed.). Springer. p. 40. ISBN 9781441960528. ^ a b Pomerance, Carl (December 1982). "The Search for Prime Numbers". Scientific American. 247 (6): 136–147. JSTOR 24966751. ^ a b c Mollin, Richard A. (2002). "A brief history of factoring and primality testing B. C. (before computers)". Mathematics Magazine. 75 (1): 18–29. doi:10.2307/3219180. MR 2107288. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics archive, University of St Andrews . ^ Sandifer 2007, 8. Fermat's Little Theorem (November 2003), p. 45 ^ Sandifer, C. Edward (2014). How Euler Did Even More. Mathematical Association of America. p. 42. ISBN 9780883855843. ^ Koshy, Thomas (2002). Elementary Number Theory with Applications. Academic Press. p. 369. ISBN 9780124211711. ^ Yuan, Wang (2002). Goldbach Conjecture. Series In Pure Mathematics. 4 (2nd ed.). World Scientific. p. 21. ISBN 9789814487528. ^ Narkiewicz, Wladyslaw (2000). "1.2 Sum of Reciprocals of Primes". The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics. Springer. p. 11. ISBN 9783540662891. ^ Apostol, Tom M. (2000). "A centennial history of the prime number theorem". In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. Number Theory. Trends in Mathematics. Basel: Birkhäuser. pp. 1–14. MR 1764793. ^ Apostol, Tom M. (1976). "7. Dirichlet's Theorem on Primes in Arithmetical Progressions". Introduction to Analytic Number Theory. New York and Heidelberg: Springer-Verlag. pp. 146–156. MR 0434929. ^ Chabert, Jean-Luc (2012). A History of Algorithms: From the Pebble to the Microchip. Springer. p. 261. ISBN 9783642181924. ^ Rosen, Kenneth H. (2000). "Theorem 9.20. Proth's Primality Test". Elementary Number Theory and Its Applications (4th ed.). Addison-Wesley. p. 342. ISBN 9780201870732. ^ Cooper, S. Barry; Hodges, Andrew (2016). The Once and Future Turing. Cambridge University Press. pp. 37–38. ISBN 9781107010833. ^ Rosen 2000, p. 245. ^ Beiler, Albert H. (1999) [1966]. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover. p. 2. ISBN 9780486210964. OCLC 444171535. ^ Katz, Shaul (2004). "Berlin roots—Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem". Science in Context. 17 (1-2): 199–234. doi:10.1017/S0269889704000092. MR 2089305. ^ a b c Kraft, James S.; Washington, Lawrence C. (2014). Elementary Number Theory. Textbooks in mathematics. CRC Press. p. 7. ISBN 9781498702690. ^ Bauer, Craig P. (2013). Secret History: The Story of Cryptology. Discrete Mathematics and Its Applications. CRC Press. p. 468. ISBN 9781466561861. ^ Klee, Victor; Wagon, Stan (1991). Old and New Unsolved Problems in Plane Geometry and Number Theory. Dolciani mathematical expositions. 11. Cambridge University Press. p. 224. ISBN 9780883853153. ^ a b Neale 2017, pp. 18 and 47. ^ a b Caldwell, Chris K.; Reddick, Angela; Xiong, Yeng; Keller, Wilfrid (2012). "The history of the primality of one: a selection of sources". Journal of Integer Sequences. 15 (9): Article 12.9.8. MR 3005523. For a selection of quotes from and about the ancient Greek positions on this issue, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6. ^ Tarán, Leonardo (1981). Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary. Philosophia Antiqua : A Series of Monographs on Ancient Philosophy. 39. BRILL. pp. 35–38. ISBN 9789004065055. ^ Caldwell et al. 2012, pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet. ^ Caldwell et al. 2012, p. 15. ^ a b c Caldwell, Chris K.; Xiong, Yeng (2012). "What is the smallest prime?" (PDF). Journal of Integer Sequences. 15 (9): Article 12.9.7. MR 3005530. ^ Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization (2nd ed.). Basel, Switzerland: Birkhäuser. p. 36. ISBN 978-0-8176-3743-9. MR 1292250. ^ a b Conway, John Horton; Guy, Richard K. (1996). The Book of Numbers. New York: Copernicus. pp. 129–130. ISBN 978-0-387-97993-9. MR 1411676. ^ For the totient, see Sierpiński 1988, p. 245. For the sum of divisors, see Sandifer, C. Edward (2007). How Euler Did It. MAA Spectrum. Mathematical Association of America. p. 59. ISBN 9780883855638. ^ Smith, Karl J. (2011). The Nature of Mathematics (12th ed.). Cengage Learning. p. 188. ISBN 9780538737586. ^ Dudley 1978, Section 2, Theorem 2, p. 16; Neale, Vicky (2017). Closing the Gap: The Quest to Understand Prime Numbers. Oxford University Press. p. 107. ISBN 9780191092435. ^ du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. p. 23. ISBN 9780060935580. ^ Dudley 1978, Section 2, Lemma 5, p. 15; Higgins, Peter M. (1998). Mathematics for the Curious. Oxford University Press. pp. 77–78. ISBN 9780191500503. ^ Rotman, Joseph J. (2000). A First Course in Abstract Algebra (2nd ed.). Prentice Hall. Problem 1.40, p. 56. ISBN 9780130115843. ^ Letter in Latin from Goldbach to Euler, July 1730. ^ Furstenberg, Harry (1955). "On the infinitude of primes". American Mathematical Monthly. Mathematical Association of America. 62 (5): 353. doi:10.2307/2307043. MR 0068566. ^ Ribenboim, Paulo (2004). The little book of bigger primes. Berlin, New York: Springer-Verlag. p. 4. ISBN 978-0-387-20169-6. ^ Euclid's Elements, Book IX, Proposition 20. See David Joyce's English translation of Euclid's proof or Williamson, James (1782). The Elements of Euclid, With Dissertations. Oxford: Clarendon Press. p. 63. OCLC 642232959. ^ Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. pp. 82–89. ISBN 9780201529890. ^ a b c Matiyasevich, Yuri V. (1999). "Formulas for prime numbers". In Tabachnikov, Serge. Kvant Selecta: Algebra and Analysis,. II. American Mathematical Society. pp. 13–24. ISBN 978-0-8218-1915-9. ^ Mackinnon, Nick (June 1987). "Prime number formulae". The Mathematical Gazette. 71 (456): 113. doi:10.2307/3616496. ^ Wright, E. M. (1951). "A prime-representing function". American Mathematical Monthly. 58 (9): 616–618. doi:10.2307/2306356. JSTOR 2306356. ^ Guy 2013, p. vii. ^ Guy 2013, C1 Goldbach's conjecture, pp. 105–107. ^ Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio (2014). "Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4 ⋅ 10 18 {\displaystyle 4\cdot 10^{18}} ". Mathematics of Computation. 83 (288): 2033–2060. doi:10.1090/S0025-5718-2013-02787-1. MR 3194140. ^ Tao 2009, 3.1 Structure and randomness in the prime numbers, pp. 239–247. See especially p. 239. ^ Guy 2013, p. 159. ^ Ramaré, Olivier (1995). "On Šnirel'man's constant". Annali della Scuola Normale Superiore di Pisa. 22 (4): 645–706. MR 1375315. ^ Rassias, Michael Th. (2017). Goldbach's Problem: Selected Topics. Cham: Springer. p. vii. doi:10.1007/978-3-319-57914-6. ISBN 978-3-319-57912-2. MR 3674356. ^ Koshy 2002, Theorem 2.14, p. 109. Riesel 1994 gives a similar argument using the primorial in place of the factorial. ^ a b Riesel 1994, "Large gaps between consecutive primes", pp. 78–79. ^ "Sloane's A100964 : Smallest prime number that begins a prime gap of at least 2n". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. ^ a b c Ribenboim 2004, Gaps between primes, pp. 186–192. ^ a b Ribenboim 2004, p. 183. ^ Chan, Joel (February 1996). "Prime time!". Math Horizons. 3 (3): 23–25. JSTOR 25678057. Note that Chan lists Legendre's conjecture as "Sierpinski's Postulate". ^ Ribenboim 2004, Prime k {\displaystyle k} -tuples conjecture, pp. 201–202. ^ Sandifer 2007, Chapter 35, Estimating the Basel problem, pp. 205–208. ^ Ogilvy, C. S.; Anderson, J. T. (1988). Excursions in Number Theory. Dover Publications Inc. pp. 29–35. ISBN 0-486-25778-9. ^ Apostol 1976, Section 1.6, Theorem 1.13 ^ Apostol 1976, Section 4.8, Theorem 4.12 ^ a b Miller, Steven J.; Takloo-Bighash, Ramin (2006). An Invitation to Modern Number Theory. Princeton University Press. pp. 43–44. ISBN 9780691120607. ^ Crandall & Pomerance 2005, p. 6. ^ Crandall & Pomerance 2005, Section 3.7, Counting primes, pp. 152–162. ^ a b Crandall & Pomerance 2005, p. 10. ^ du Sautoy, Marcus (2011). "What are the odds that your telephone number is prime?". The Number Mysteries: A Mathematical Odyssey through Everyday Life. St. Martin's Press. pp. 50–52. ISBN 9780230120280. ^ Apostol 1976, Section 4.6, Theorem 4.7 ^ Gelfand, I. M.; Shen, Alexander (2003). Algebra. Springer. p. 37. ISBN 9780817636777. ^ Mollin, Richard A. (1997). Fundamental Number Theory with Applications. Discrete Mathematics and Its Applications. CRC Press. p. 76. ISBN 9780849339875. ^ Crandall & Pomerance 2005, Theorem 1.1.5, p. 12. ^ Green, Ben; Tao, Terence (2008). "The primes contain arbitrarily long arithmetic progressions". Annals of Mathematics. 167 (2): 481–547. arXiv:math.NT/0404188 . doi:10.4007/annals.2008.167.481. ^ Hua, L. K. (2009) [1965]. Additive Theory of Prime Numbers. Translations of Mathematical Monographs. 13. Providence, RI: American Mathematical Society. pp. 176–177. ISBN 978-0-8218-4942-2. MR 0194404. OCLC 824812353. ^ The sequence of these primes, starting at n = 1 {\displaystyle n=1} rather than n = 0 {\displaystyle n=0} , is listed by Lava, Paolo Pietro; Balzarotti, Giorgio (2010). "Chapter 33. Formule fortunate". 103 curiosità matematiche: Teoria dei numeri, delle cifre e delle relazioni nella matematica contemporanea (in Italian). Ulrico Hoepli Editore S.p.A. p. 133. ISBN 9788820358044. ^ Chamberland, Marc (2015). "The Heegner numbers". Single Digits: In Praise of Small Numbers. Princeton University Press. pp. 213–215. ISBN 9781400865697. ^ a b Guy, Richard (2013). "A1 Prime values of quadratic functions". Unsolved Problems in Number Theory. Problem Books in Mathematics (3rd ed.). Springer. pp. 7–10. ISBN 9780387266770. ^ Patterson, S. J. (1988). An introduction to the theory of the Riemann zeta-function. Cambridge Studies in Advanced Mathematics. 14. Cambridge University Press, Cambridge. p. 1. doi:10.1017/CBO9780511623707. ISBN 0-521-33535-3. MR 0933558. ^ Borwein, Peter; Choi, Stephen; Rooney, Brendan; Weirathmueller, Andrea (2008). The Riemann hypothesis: A resource for the afficionado and virtuoso alike. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York: Springer. pp. 10–11. doi:10.1007/978-0-387-72126-2. ISBN 978-0-387-72125-5. MR 2463715. ^ Sandifer 2007, pp. 191–193. ^ Borwein et al. 2008, Conjecture 2.7 (the Riemann hypothesis), p. 15. ^ Patterson 1988, p. 7. ^ a b Borwein et al. 2008, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA18 p. 18. ^ Nathanson 2000, Chapter 9, The prime number theorem, pp. 289–324. ^ Zagier, Don (1977). "The first 50 million prime numbers". The Mathematical Intelligencer. 1 (S2): 7–19. doi:10.1007/bf03351556. See especially pp. 14–16. ^ Kraft & Washington (2014), Proposition 5.3, p. 96. ^ Shahriari, Shahriar (2017). Algebra in Action: A Course in Groups, Rings, and Fields. Pure and Applied Undergraduate Texts. 27. American Mathematical Society. pp. 20–21. ISBN 9781470428495. ^ Dudley 1978, Theorem 3, p. 28. ^ Shahriari 2017, pp. 27–28. ^ Ribenboim 2004, Fermat's little theorem and primitive roots modulo a prime, pp. 17–21. ^ Ribenboim 2004, The property of Giuga, pp. 21–22. ^ Ribenboim 2004, The theorem of Wilson, p. 21. ^ a b c Childress, Nancy (2009), Class Field Theory, Universitext, Springer, New York, pp. 8–11, doi:10.1007/978-0-387-72490-4, ISBN 978-0-387-72489-8, MR 2462595 See also p. 64. ^ Erickson, Marty; Vazzana, Anthony; Garth, David (2016). Introduction to Number Theory. Textbooks in Mathematics (2nd ed.). Boca Raton, FL: CRC Press. p. 200. ISBN 978-1-4987-1749-6. MR 3468748. ^ Weil, André (1995). Basic Number Theory. Classics in Mathematics. Berlin: Springer-Verlag. p. 43. ISBN 3-540-58655-5. MR 1344916. Note however that some authors such as Childress (2009) instead use "place" to mean an equivalence class of norms. ^ Koch, H. (1997). Algebraic Number Theory. Berlin: Springer-Verlag. p. 136. doi:10.1007/978-3-642-58095-6. ISBN 3-540-63003-1. MR 1474965. ^ Lauritzen, Niels (2003). Concrete Abstract Algebra: From numbers to Gröbner bases. Cambridge: Cambridge University Press. p. 127. doi:10.1017/CBO9780511804229. ISBN 0-521-53410-0. MR 2014325. ^ Lauritzen 2003, Corollary 3.5.14, p. 133; Lemma 3.5.18, p. 136. ^ Kraft & Washington 2014, Section 12.1, Sums of two squares, pp. 297–301. ^ Eisenbud, David (1995). Commutative Algebra. Graduate Texts in Mathematics. 150. Berlin, New York: Springer-Verlag. Section 3.3. ISBN 978-0-387-94268-1. MR 1322960. ^ Shafarevich, Igor R. (2013). "Definition of Spec ⁡ A {\displaystyle \operatorname {Spec} A} ". Basic Algebraic Geometry 2: Schemes and Complex Manifolds (3rd ed.). Springer, Heidelberg. p. 5. doi:10.1007/978-3-642-38010-5. ISBN 978-3-642-38009-9. MR 3100288. ^ Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 322. Berlin: Springer-Verlag. Section I.8, p. 50. doi:10.1007/978-3-662-03983-0. ISBN 3-540-65399-6. MR 1697859. ^ Neukirch 1999, Section I.7, p. 38 ^ Stevenhagen, P.; Lenstra, H. W., Jr. (1996). "Chebotarëv and his density theorem". The Mathematical Intelligencer. 18 (2): 26–37. doi:10.1007/BF03027290. MR 1395088. ^ Hall, Marshall (2018). The Theory of Groups. Dover Books on Mathematics. Courier Dover Publications. ISBN 9780486816906. For the Sylow theorems see p. 43; for Lagrange's theorem, see p. 12; for the Burnside theorem see p. 143. ^ Bryant, John; Sangwin, Christopher J. (2008). How Round is Your Circle?: Where Engineering and Mathematics Meet. Princeton University Press. p. 178. ISBN 9780691131184. ^ Hardy, Godfrey Harold (2012) [1940]. A Mathematician's Apology. Cambridge University Press. p. 140. ISBN 978-0-521-42706-7. OCLC 922010634. No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years. ^ Giblin, P. J. (1993). Primes and Programming. Cambridge University Press. p. 39. ISBN 9780521409889. ^ Giblin 1993, p. 54 ^ Riesel 1994, p. 220. ^ Bullynck, Maarten (2010). "A history of factor tables with notes on the birth of number theory 1657–1817". Revue d'Histoire des Mathématiques. 16 (2): 133–216. ^ Wagstaff, Samuel S. Jr. (2013). The Joy of Factoring. Student mathematical library. 68. American Mathematical Society. p. 191. ISBN 9781470410483. ^ Crandall, Richard; Pomerance, Carl (2005). Prime Numbers: A Computational Perspective (2nd ed.). Springer. p. 121. ISBN 9780387252827. ^ Farach-Colton, Martín; Tsai, Meng-Tsung (2015). "On the complexity of computing prime tables". In Elbassioni, Khaled; Makino, Kazuhisa. Algorithms and Computation: 26th International Symposium, ISAAC 2015, Nagoya, Japan, December 9-11, 2015, Proceedings. Lecture Notes in Computer Science. 9472. Springer. pp. 677–688. arXiv:1504.05240 . doi:10.1007/978-3-662-48971-0_57. ^ Greaves, George (2013). Sieves in Number Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge). 43. Springer. p. 1. ISBN 9783662046586. ^ a b Hromkovič, Juraj (2001). "5.5 Bibliographic Remarks". Algorithmics for Hard Problems. Texts in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin. pp. 383–385. doi:10.1007/978-3-662-04616-6. ISBN 3-540-66860-8. MR 1843669. ^ a b Koblitz, Neal (1987). "Chapter V. Primality and Factoring". A Course in Number Theory and Cryptography. Graduate Texts in Mathematics. 114. Springer-Verlag, New York. pp. 112–149. doi:10.1007/978-1-4684-0310-7_5. ISBN 0-387-96576-9. MR 0910297. ^ Pieprzyk, Josef; Hardjono, Thomas; Seberry, Jennifer (2013). "2.3.9 Probabilistic Computations". Fundamentals of Computer Security. Springer. pp. 51–52. ISBN 9783662073247. ^ a b Tao, Terence (2010). "1.11 The AKS primality test". An epsilon of room, II: Pages from year three of a mathematical blog. Providence, RI: American Mathematical Society. pp. 82–86. doi:10.1090/gsm/117. ISBN 978-0-8218-5280-4. MR 2780010. ^ a b Morain, F. (2007). "Implementing the asymptotically fast version of the elliptic curve primality proving algorithm". Mathematics of Computation. 76 (257): 493–505. doi:10.1090/S0025-5718-06-01890-4. MR 2261033. ^ Atkin, A. O. L.; Morain, F. (1993). "Elliptic curves and primality proving". Mathematics of Computation. 61 (203): 29–68. doi:10.2307/2152935. MR 1199989. ^ Lenstra, Jr., H. W.; Pomerance, Carl (December 11, 2016). "Primality testing with Gaussian periods" (PDF). ^ a b Monier, Louis (1980). "Evaluation and comparison of two efficient probabilistic primality testing algorithms". Theoretical Computer Science. 12 (1): 97–108. doi:10.1016/0304-3975(80)90007-9. MR 0582244. ^ Tao, Terence (2009). "1.7 The Lucas–Lehmer test for Mersenne primes". Poincaré's legacies, pages from year two of a mathematical blog. Part I. Providence, RI: American Mathematical Society. pp. 36–41. ISBN 978-0-8218-4883-8. MR 2523047. ^ Kraft & Washington 2014, p. 41. ^ For instance see Guy 2013, A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape k ⋅ 2 n + 1 {\displaystyle k\cdot 2^{n}+1} . pp. 13–21. ^ "Record 12-Million-Digit Prime Number Nets$100,000 Prize". Electronic Frontier Foundation. October 14, 2009. Retrieved 2010-01-04.  ^ "EFF Cooperative Computing Awards". Electronic Frontier Foundation. Retrieved 2010-01-04.  ^ Priday, Richard (January 5, 2018). "GIMPS crack whip on plucky processor to find largest prime number: 23 million-digit figure revealed after six days of non-stop calculations". The Register. Retrieved February 22, 2018.  ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). Retrieved 2017-01-03.  ^ Caldwell, Chris K. "The Top Twenty: Largest Known Primes". The Prime Pages. Retrieved 2017-01-03.  ^ Caldwell, Chris K. "The Top Twenty: Factorial". The Prime Pages. Retrieved 2017-01-03.  ^ Ribenboim 2004, p. 4. ^ Caldwell, Chris K. "The Top Twenty: Primorial". The Prime Pages. Retrieved 2017-01-03.  ^ Caldwell, Chris K. "The Top Twenty: Twin Primes". The Prime Pages. Retrieved 2017-01-03.  ^ Kraft & Washington 2014, p. 275. ^ Riesel 1994, p. 220. ^ Hoffstein, Jeffrey; Pipher, Jill; Silverman, Joseph H. (2014). An Introduction to Mathematical Cryptography. Undergraduate Texts in Mathematics (2nd ed.). Springer. p. 329. ISBN 9781493917112.  ^ Pomerance, Carl (1996). "A tale of two sieves". Notices of the American Mathematical Society. 43 (12): 1473–1485. MR 1416721.  ^ Kleinjung, T.; Aoki, K.; Franke, J.; Lenstra, A. K.; Thomé, E.; Bos, J. W.; Gaudry, P.; Kruppa, A.; Montgomery, P. L.; Osvik, D. A.; te Riele, H.; Timofeev, A.; Zimmermann, P. "Factorization of a 768-bit RSA modulus" (PDF). Retrieved 2013-04-11.  ^ Rieffel, Eleanor G.; Polak, Wolfgang H. (2011). "Chapter 8. Shor's Algorithm". Quantum Computing: A Gentle Introduction. MIT Press. pp. 163–176. ISBN 9780262015066.  ^ Martín-López, Enrique; Laing, Anthony; Lawson, Thomas; Alvarez, Roberto; Zhou, Xiao-Qi; O'Brien, Jeremy L. (12 October 2012). "Experimental realization of Shor's quantum factoring algorithm using qubit recycling". Nature Photonics. 6 (11): 773. arXiv:1111.4147 . Bibcode:2012NaPho...6..773M. doi:10.1038/nphoton.2012.259.  ^ Chirgwin, Richard (October 9, 2016). "Crypto needs more transparency, researchers warn". The Register.  ^ Hoffstein, Pipher & Silverman 2014, Section 2.3, Diffie–Hellman key exchange, pp. 65–67. ^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990]. "11.3 Universal hashing". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 232–236. ISBN 0-262-03293-7.  For k {\displaystyle k} -independent hashing see problem 11–4, p. 251. For the credit to Carter and Wegman, see the chapter notes, p. 252. ^ Goodrich, Michael T.; Tamassia, Roberto (2006). Data Structures & Algorithms in Java (4th ed.). John Wiley & Sons. ISBN 9780471738848.  See "Quadratic probing", p. 382, and exercise C–9.9, p. 415. ^ Kirtland, Joseph (2001). Identification Numbers and Check Digit Schemes. Classroom Resource Materials. 18. Mathematical Association of America. pp. 43–44. ISBN 9780883857205.  ^ Deutsch, P. (1996). ZLIB Compressed Data Format Specification version 3.3. Request for Comments. 1950. Network Working Group.  ^ Knuth, Donald E. (1998). "3.2.1 The linear congruential model". The Art of Computer Programming, Vol. 2: Seminumerical algorithms (3rd ed.). Addison-Wesley. pp. 10–26. ISBN 978-0-201-89684-8.  ^ Matsumoto, Makoto; Nishimura, Takuji (1998). "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudo-random number generator". ACM Transactions on Modeling and Computer Simulation. 8 (1): 3–30. doi:10.1145/272991.272995.  ^ Roth, K. F. (1951). "On a problem of Heilbronn". Journal of the London Mathematical Society. Second Series. 26: 198–204. doi:10.1112/jlms/s1-26.3.198. MR 0041889.  ^ Cox, David A. (2011). "Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first" (PDF). American Mathematical Monthly. 118 (1): 3–31. doi:10.4169/amer.math.monthly.118.01.003.  ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. 211. Berlin, New York: Springer-Verlag. ISBN 978-0-387-95385-4. MR 1878556. , Section II.1, p. 90 ^ Schubert, Horst (1949). "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (3): 57–104. MR 0031733.  ^ Milnor, J. (1962). "A unique decomposition theorem for 3-manifolds". American Journal of Mathematics. 84: 1–7. doi:10.2307/2372800. MR 0142125.  ^ Boklan & Conway (2017) also include 2 0 + 1 = 2 {\displaystyle 2^{0}+1=2} , which is not of this form. ^ a b Křížek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics. 9. New York: Springer-Verlag. pp. 1–2. doi:10.1007/978-0-387-21850-2. ISBN 0-387-95332-9. MR 1866957.  ^ Boklan, Kent D.; Conway, John H. (January 2017). "Expect at most one billionth of a new Fermat prime!". The Mathematical Intelligencer. 39 (1): 3–5. arXiv:1605.01371 . doi:10.1007/s00283-016-9644-3.  ^ Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly. 95 (3): 185–194. doi:10.2307/2323624. MR 0935432.  ^ Ziegler, Günter M. (2015). "Cannons at sparrows". European Mathematical Society (95): 25–31. MR 3330472.  ^ Peterson, Ivars (June 28, 1999). "The Return of Zeta". MAA Online. Archived from the original on October 20, 2007. Retrieved 2008-03-14.  ^ Hayes, Brian (2003). "Computing science: The spectrum of Riemannium". American Scientist. 91 (4): 296–300. JSTOR 27858239.  ^ Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of quantum states : an introduction to quantum entanglement (Second ed.). Cambridge, United Kingdom: Cambridge University Press. pp. 313–354. ISBN 9781107026254. OCLC 967938939.  ^ Zhu, Huangjun. "SIC POVMs and Clifford groups in prime dimensions". Journal of Physics A: Mathematical and Theoretical. 43 (30). arXiv:1003.3591 . doi:10.1088/1751-8113/43/30/305305.  ^ Goles, E.; Schulz, O.; Markus, M. (2001). "Prime number selection of cycles in a predator-prey model". Complexity. 6 (4): 33–38. doi:10.1002/cplx.1040.  ^ Campos, Paulo R. A.; de Oliveira, Viviane M.; Giro, Ronaldo; Galvão, Douglas S. (2004). "Emergence of prime numbers as the result of evolutionary strategy". Physical Review Letters. 93 (9): 098107. arXiv:q-bio/0406017 . Bibcode:2004PhRvL..93i8107C. doi:10.1103/PhysRevLett.93.098107.  ^ "Invasion of the Brood". The Economist. May 6, 2004. Retrieved 2006-11-26.  ^ Zimmer, Carl (May 15, 2015). "Bamboo Mathematicians". Phenomena: The Loom. National Geographic. Retrieved February 22, 2018.  ^ Hill, Peter Jensen, ed. (1995). The Messiaen companion. Portland, Or: Amadeus Press. Ex. 13.2 Messe de la Pentecôte 1 'Entrée'. ISBN 978-0-931340-95-6.  ^ Pomerance, Carl (2004). "Prime Numbers and the Search for Extraterrestrial Intelligence" (PDF). In Hayes, David F.; Ross, Peter. Mathematical Adventures for Students and Amateurs. MAA Spectrum. Washington, DC: Mathematical Association of America. pp. 3–6. ISBN 0-88385-548-8. MR 2085842.  ^ GrrlScientist (September 16, 2010). "The Curious Incident of the Dog in the Night-Time". Science. The Guardian. Retrieved February 22, 2010.  ^ Schillinger, Liesl (April 9, 2010). "Counting on Each Other". Sunday Book Review. The New York Times.

External links Look up prime number in Wiktionary, the free dictionary. Wikinews has related news: Two largest known prime numbers discovered just two weeks apart; one qualifies for \$100k prize Record size 17.4 million-digit prime found Wikimedia Commons has media related to Prime number. Hazewinkel, Michiel, ed. (2001) [1994], "Prime number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4  Caldwell, Chris, The Prime Pages at primes.utm.edu. Prime Numbers on In Our Time at the BBC. Plus teacher and student package: prime numbers from Plus, the free online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge. Generators and calculators Prime Number Checker identifies the smallest prime factor of a number. Fast Online primality test with factorization makes use of the Elliptic Curve Method (up to thousand-digits numbers, requires Java). Huge database of prime numbers Prime Numbers up to 1 trillion v t e Divisibility-based sets of integers Overview Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number Factorization forms Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual Constrained divisor sums Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas With many divisors Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird Aliquot sequence-related Untouchable Amicable Sociable Betrothed Other sets Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant v t e Prime number classes By formula Fermat (22n + 1) Mersenne (2p − 1) Double Mersenne (22p−1 − 1) Wagstaff (2p + 1)/3 Proth (k·2n + 1) Factorial (n! ± 1) Primorial (pn# ± 1) Euclid (pn# + 1) Pythagorean (4n + 1) Pierpont (2u·3v + 1) Quartan (x4 + y4) Solinas (2a ± 2b ± 1) Cullen (n·2n + 1) Woodall (n·2n − 1) Cuban (x3 − y3)/(x − y) Carol (2n − 1)2 − 2 Kynea (2n + 1)2 − 2 Leyland (xy + yx) Thabit (3·2n − 1) Mills (⌊A3n⌋) By integer sequence Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin By property Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Base-dependent Happy Dihedral Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Patterns Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n) By size Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known Complex numbers Eisenstein prime Gaussian prime Composite numbers Pseudoprime Almost prime Semiprime Interprime Related topics Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap First 50 primes 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 List of prime numbers Authority control LCCN: sh85093218 GND: 4047263-2 NDL: 00571462 Retrieved from "https://en.wikipedia.org/w/index.php?title=Prime_number&oldid=827434633" Categories: Prime numbersInteger sequencesHidden categories: CS1 Italian-language sources (it)Good articlesWikipedia pages semi-protected against vandalismArticles containing potentially dated statements from January 2018All articles containing potentially dated statementsArticles containing potentially dated statements from 2014Commons category with page title different than on WikidataWikipedia articles with LCCN identifiersWikipedia articles with GND identifiersArticles containing proofs