Prime numberPrime redirects here. For other uses, seePrime(disambiguation).Aprime number(ora prime)isanatural numbergreater than 1 that has no positive divisors other than 1and itself.A natural number greater than 1 that is not aprime number is called a composite number. For exam-ple, 5 is prime because 1 and 5 are its only positive integerfactors, whereas 6 is composite because it has the divisors2 and 3 in addition to 1 and 6. The fundamental theo-rem of arithmetic establishes the central role of primesin number theory: any integer greater than 1 can be ex-pressed as a product of primes that is unique up to order-ing. The uniqueness in this theorem requires excluding 1as a prime because one can include arbitrarily many in-stances of 1 in any factorization, e.g., 3, 1 3, 1 1 3,etc. are all valid factorizations of 3.The property of being prime (or not) is called primality.A simple but slow method of verifying the primality of agiven number n is known as trial division. It consists oftesting whether n is a multiple of any integer between 2and n . Algorithms much more ecient than trial divi-sion have been devised to test the primality of large num-bers. Particularly fast methods are available for numbersof special forms, such as Mersenne numbers. As of April2014, the largest known prime number has 17,425,170decimal digits.There are innitely many primes, as demonstrated by Eu-clid around 300 BC. There is no known useful formulathat sets apart all of the prime numbers from composites.However, the distribution of primes, that is to say, the sta-tistical behaviour of primes in the large, can be modelled.The rst result in that direction is the prime number the-orem, proven at the end of the 19th century, which saysthat the probability that a given, randomly chosen num-ber n is prime is inversely proportional to its number ofdigits, or to the logarithm of n.Many questions regarding prime numbers remain open,such as Goldbachs conjecture (that every even integergreater than 2 can be expressed as the sumof two primes),and the twin prime conjecture (that there are innitelymanypairsofprimeswhosedierenceis2). Suchquestions spurred the development of various branchesof number theory, focusing on analytic or algebraic as-pects of numbers. Primes are used in several routinesin information technology, such as public-key cryptog-raphy, which makes use of properties such as the dif-culty of factoring large numbers into their prime fac-tors.Prime numbers give rise to various generalizationsin other mathematical domains, mainly algebra, such asprime elements and prime ideals.1 Denition and examplesA natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called aprime number (or a prime) if it has exactly two posi-tive divisors, 1 and the number itself.[1] Natural numbersgreater than 1 that are not prime are called composite.The number 12 is not a prime, as 12 items can be placed into3 equal-size columns of 4 each (among other ways). 11 itemscannot beall placedintoseveral equal-sizecolumnsofmorethan 1 itemeach without some extra items leftover (a remainder).Therefore the number 11 is a prime.Among the numbers 1 to 6, the numbers 2, 3, and 5 arethe prime numbers, while 1, 4, and 6 are not prime. 1is excluded as a prime number, for reasons explained be-low. 2 is a prime number, since the only natural numbersdividing it are 1 and 2. Next, 3 is prime, too: 1 and 3do divide 3 without remainder, but 3 divided by 2 givesremainder 1. Thus, 3 is prime. However, 4 is composite,since 2 is another number (in addition to 1 and 4) dividing4 without remainder:4 = 2 2.5 is again prime: none of the numbers 2, 3, or 4 divide 5.Next, 6 is divisible by 2 or 3, since6 = 2 3.12 2 FUNDAMENTAL THEOREM OF ARITHMETICHence, 6 is not prime.The image at the right illustratesthat 12 is not prime: 12 = 3 4. No even number greaterthan 2 is prime because by denition, any such numbern has at least three distinct divisors, namely 1, 2, andn. This implies thatn is not prime. Accordingly, theterm odd prime refers to any prime number greater than2. Similarly, when written in the usual decimal system,all prime numbers larger than 5 end in 1, 3, 7, or 9, sinceeven numbers are multiples of 2 and numbers ending in0 or 5 are multiples of 5.If n is a natural number, then 1 and n divide n withoutremainder. Therefore, the condition of being a prime canalso be restated as: a number is prime if it is greater thanone and if none of2, 3, ..., n 1divides n (without remainder). Yet another way to say thesame is: a number n > 1 is prime if it cannot be writtenas a product of two integers a and b, both of which arelarger than 1:n = a b.In other words, n is prime if n items cannot be divided upinto smaller equal-size groups of more than one item.The set of all primes is often denoted by P.The rst 168 prime numbers (all the prime numbers lessthan 1000) are: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, 233, 239, 241,251, 257, 263, 269, 271, 277, 281, 283, 293,307, 311, 313, 317, 331, 337, 347, 349, 353,359, 367, 373, 379, 383, 389, 397, 401, 409,419, 421, 431, 433, 439, 443, 449, 457, 461,463, 467, 479, 487, 491, 499, 503, 509, 521,523, 541, 547, 557, 563, 569, 571, 577, 587,593, 599, 601, 607, 613, 617, 619, 631, 641,643, 647, 653, 659, 661, 673, 677, 683, 691,701, 709, 719, 727, 733, 739, 743, 751, 757,761, 769, 773, 787, 797, 809, 811, 821, 823,827, 829, 839, 853, 857, 859, 863, 877, 881,883, 887, 907, 911, 919, 929, 937, 941, 947,953, 967, 971, 977, 983, 991, 997 (sequenceA000040 in OEIS).2 Fundamental theoremof arith-meticMain article: Fundamental theorem of arithmeticThe crucial importance of prime numbers to number the-ory and mathematics in general stems from the funda-mental theorem of arithmetic, which states that every in-teger larger than 1 can be written as a product of one ormore primes in a way that is unique except for the orderof the prime factors.[2] Primes can thus be considered thebasic building blocks of the natural numbers. For ex-ample:As in this example, the same prime factor may occur mul-tiple times. A decomposition:n = p1 p2 ... ptof a number n into (nitely many) prime factors p1, p2, ...to pt is called prime factorization of n. The fundamentaltheorem of arithmetic can be rephrased so as to say thatany factorization into primes will be identical except forthe order of the factors. So, albeit there are many primefactorization algorithms to do this in practice for largernumbers, they all have to yield the same result.If p is a prime number and p divides a product ab of in-tegers, then p divides a or p divides b. This propositionis known as Euclids lemma.[3] It is used in some proofsof the uniqueness of prime factorizations.2.1 Primality of oneMost earlyGreeks didnot evenconsider 1tobeanumber,[4] so they could not consider it to be a prime. Bythe Middle Ages and Renaissance many mathematiciansincluded 1 as the rst prime number.[5] In the mid-18thcentury Christian Goldbach listed 1 as the rst prime inhis famous correspondence with Leonhard Euler -- whodid not agree.[6] In the 19th century many mathemati-cians still considered the number 1 to be a prime. Forexample, Derrick Norman Lehmer's list of primes up to10,006,721, reprinted as late as 1956,[7] started with 1 asits rst prime.[8] Henri Lebesgue is said to be the last pro-fessional mathematician to call 1 prime.[9] By the early20th century, mathematicians began to accept that 1 isnot a prime number, but rather forms its own special cat-egory as a unit.[10]A large body of mathematical work would still be validwhen calling 1 a prime, but Euclids fundamental theo-rem of arithmetic (mentioned above) would not hold asstated. For example, the number 15 can be factored as3 5 and 1 3 5; if 1 were admitted as a prime, thesetwo presentations would be considered dierent factor-izations of 15 into prime numbers, so the statement ofthat theorem would have to be modied. Similarly, thesieve of Eratosthenes would not work correctly if 1 wereconsidered a prime: a modied version of the sieve thatconsiders 1 as prime would eliminate all multiples of 13(that is, all other numbers) and produce as output onlythe single number 1. Furthermore, the prime numbershave several properties that the number 1 lacks, such asthe relationship of the number to its corresponding valueof Eulers totient function or the sum of divisors func-tion.[11][12]3 HistoryThe Sieve of Eratosthenes is a simple algorithm for nding allprime numbers up to a specied integer. It was created in the 3rdcentury BC by Eratosthenes, an ancient Greek mathematician.There are hints in the surviving records of the ancientEgyptians that they had some knowledge of prime num-bers:the Egyptian fraction expansions in the Rhind pa-pyrus, for instance, have quite dierent forms for primesandfor composites. However, theearliest survivingrecords of the explicit study of prime numbers comefrom the Ancient Greeks. Euclids Elements (circa 300BC) contain important theorems about primes, includingthe innitude of primes and the fundamental theorem ofarithmetic. Euclid also showed how to construct a perfectnumber from a Mersenne prime. The Sieve of Eratos-thenes, attributed to Eratosthenes, is a simple method tocompute primes, although the large primes found todaywith computers are not generated this way.After the Greeks, little happened with the study of primenumbers until the 17th century. In 1640 Pierre de Fer-mat stated (without proof) Fermats little theorem (laterproved by Leibniz and Euler). Fermat also conjecturedthat all numbers of the form 22n+ 1 are prime (they arecalled Fermat numbers) and he veried this up to n = 4 (or216+ 1). However, the very next Fermat number 232+ 1is composite (one of its prime factors is 641), as Euler dis-covered later, and in fact no further Fermat numbers areknown to be prime. The French monk Marin Mersennelooked at primes of the form 2p 1, with p a prime. Theyare called Mersenne primes in his honor.Eulers work in number theory included many resultsabout primes.He showed the innite series 1/2 + 1/3 +1/5 + 1/7 + 1/11 + is divergent. In 1747 he showed thatthe even perfect numbers are precisely the integers of theform 2p1(2p 1), where the second factor is a Mersenneprime.At the start of the 19th century, Legendre and Gauss in-dependently conjectured that as x tends to innity, thenumber of primes up to x is asymptotic to x/ln(x), whereln(x) is the natural logarithm of x. Ideas of Riemann inhis 1859 paper on the zeta-function sketched a programthat would lead to a proof of the prime number theorem.This outline was completed by Hadamard and de la Val-le Poussin, who independently proved the prime numbertheorem in 1896.Proving a number is prime is not done (for large numbers)by trial division. Many mathematicians have worked onprimality tests for large numbers, often restricted to spe-cic number forms. This includes Ppins test for Fer-mat numbers (1877), Proths theorem (around 1878), theLucasLehmer primality test (originated 1856),[13] andthe generalized Lucas primality test. More recent algo-rithms like APRT-CL, ECPP, and AKS work on arbi-trary numbers but remain much slower.For a long time, prime numbers were thought to haveextremely limited application outside of pure mathemat-ics.[14] This changed in the 1970s when the concepts ofpublic-key cryptography were invented, in which primenumbers formed the basis of the rst algorithms such asthe RSA cryptosystem algorithm.Since 1951 all the largest known primes have been foundby computers. The search for ever larger primes has gen-erated interest outside mathematical circles. The GreatInternet Mersenne Prime Search and other distributedcomputing projects to nd large primes have become pop-ular, while mathematicians continue to struggle with thetheory of primes.4 Number of prime numbersMain article: Euclids theoremThere are innitely many prime numbers. Another wayof saying this is that the sequence2, 3, 5, 7, 11, 13, ...of prime numbers never ends. This statement is referredto as Euclids theoremin honor of the ancient Greek math-ematician Euclid, since the rst known proof for thisstatement is attributed to him. Many more proofs of theinnitude of primes are known, including an analyticalproof by Euler, Goldbachs proof based on Fermat num-bers,[15] Furstenbergs proof using general topology,[16]and Kummers elegant proof.[17]4 5 TESTING PRIMALITY AND INTEGER FACTORIZATION4.1 Euclids proofEuclids proof (Book IX, Proposition 20[18]) considersany nite set S of primes. The key idea is to considerthe product of all these numbers plus one:N= 1 +pSp.Like any other natural number, N is divisible by at leastone prime number (it is possible that N itself is prime).None of the primes by which N is divisible can be mem-bers of the nite set S of primes with which we started,because dividing N by any one of these leaves a remain-der of 1. Therefore the primes by which N is divisible areadditional primes beyond the ones we started with. Thusany nite set of primes can be extended to a larger niteset of primes.It is often erroneously reported that Euclid begins withthe assumption that the set initially considered containsall prime numbers, leading to a contradiction, or that itcontains precisely the n smallest primes rather than anyarbitrary nite set of primes.[19] Today, the product ofthe smallest n primes plus 1 is conventionally called thenth Euclid number.4.2 Eulers analytical proofEulers proof uses the sum of the reciprocals of primes,S(p) =12+13+15+17+ +1p.This sum becomes larger than any arbitrary real num-ber provided that p is big enough.[20]This shows thatthere are innitely many primes, since otherwise this sumwould grow only until the biggest primep is reached.The growth of S(p) is quantied by Mertens second the-orem.[21] For comparison, the sum112+122+132+ +1n2=ni=11i2does not grow to innity as n goes to innity (see Baselproblem). In this sense, prime numbers occur more oftenthan squares of natural numbers. Bruns theorem statesthat the sum of the reciprocals of twin primes,(13+15)+(15+17)+(111+113)+ =p prime,p+2prime(1p+1p + 2),is nite.5 Testing primality and integerfactorizationThere are various methods to determine whether a givennumber n is prime. The most basic routine, trial divi-sion, is of little practical use because of its slowness. Onegroup of modern primality tests is applicable to arbitrarynumbers, while more ecient tests are available for par-ticular numbers. Most such methods only tell whether nis prime or not. Routines also yielding one (or all) primefactors of n are called factorization algorithms.5.1 Trial divisionThe most basic method of checking the primality of agiven integer n is called trial division. This routine con-sists of dividing n by each integer m that is greater than1 and less than or equal to the square root of n. If theresult of any of these divisions is an integer, then n is nota prime, otherwise it is a prime. Indeed, ifn=ab iscomposite (with a and b 1) then one of the factors a orb is necessarily at most n . For example, for n=37 ,the trial divisions are by m = 2, 3, 4, 5, and 6. None ofthese numbers divides 37, so 37 is prime. This routinecan be implemented more eciently if a complete list ofprimes up to n is knownthen trial divisions need tobe checked only for those m that are prime. For example,to check the primality of 37, only three divisions are nec-essary (m = 2, 3, and 5), given that 4 and 6 are composite.While a simple method, trial division quickly becomesimpractical for testing large integers because the numberof possible factors grows too rapidly as n increases. Ac-cording to the prime number theorem explained below,the number of prime numbers less than n is approxi-mately given by n/ ln(n) , so the algorithm may needup to this number of trial divisions to check the primalityof n. For n = 1020, this number is 450 milliontoo largefor many practical applications.5.2 SievesAn algorithm yielding all primes up to a given limit, suchas required in the primes-only trial division method, iscalled a prime number sieve. The oldest example, thesieve of Eratosthenes (see above), is still the most com-monly used. The sieve of Atkin is another option. Beforethe advent of computers, lists of primes up to bounds like107were also used.[22]5.3 Primality testing versus primalityprovingModern primality tests for generalnumbersn can bedivided into two main classes, probabilistic (or Monte5.4 Special-purpose algorithms and the largest known prime 5Carlo) and deterministic algorithms. Deterministic al-gorithms provide a way to tell for sure whether a givennumber is prime or not. For example, trial division isa deterministic algorithm because, if it performed cor-rectly, it will always identify a prime number as primeand a composite number as composite. Probabilistic al-gorithms are normally faster, but do not completely provethat a number is prime. These tests rely on testing a givennumber in a partly random way. For example, a giventest might pass all the time if applied to a prime number,but pass only with probability p if applied to a compos-ite number.If we repeat the test n times and pass everytime, then the probability that our number is compositeis 1/(1-p)n, which decreases exponentially with the num-ber of tests, so we can be as sure as we like (though neverperfectly sure) that the number is prime. On the otherhand, if the test ever fails, then we know that the numberis composite.Aparticularly simple example of a probabilistic test is theFermat primality test, which relies on the fact (Fermatslittle theorem) that npn (mod p) for any n if p is a primenumber.If we have a number b that we want to test forprimality, then we work out nb(mod b) for a randomvalueof n as our test. Aawwith this test is that there are somecomposite numbers (the Carmichael numbers) that sat-isfy the Fermat identity even though they are not prime, sothe test has no way of distinguishing between prime num-bers and Carmichael numbers. Carmichael numbers aresubstantially rarer than prime numbers, though, so thistest can be useful for practical purposes. More power-ful extensions of the Fermat primality test, such as theBaillie-PSW, Miller-Rabin, and Solovay-Strassen tests,are guaranteed to fail at least some of the time when ap-plied to a composite number.Deterministic algorithms do not erroneously report com-posite numbers as prime. In practice, the fastest suchmethod is known as elliptic curve primality proving. An-alyzing its run time is based on heuristic arguments, asopposed to the rigorously proven complexity of the morerecent AKS primality test. Deterministic methods aretypically slower than probabilistic ones, so the latter onesare typically applied rst before a more time-consumingdeterministic routine is employed.The following table lists a number of prime tests. Therunning time is given in terms of n, the number to betested and, for probabilistic algorithms, the number k oftests performed. Moreover, is an arbitrarily small pos-itive number, and log is the logarithm to an unspeciedbase. The big Onotation means that, for example, ellipticcurve primality proving requires a time that is bounded bya factor (not depending on n, but on ) times log5+(n).5.4 Special-purpose algorithms and thelargest known primeFurther information: List of prime numbersIn addition to the aforementioned tests applying to anyConstruction of a regular pentagon. 5 is a Fermat prime.natural number n, a number of much more ecient pri-mality tests is available for special numbers. For exam-ple, to run Lucas primality test requires the knowledgeof the prime factors of n 1, while the LucasLehmerprimality test needs the prime factors of n + 1 as input.For example, these tests can be applied to check whethern! 1 = 1 2 3 ... n 1are prime. Prime numbers of this form are known asfactorial primes. Other primes where eitherp + 1 orp 1 is of a particular shape include the Sophie Ger-main primes (primes of the form 2p + 1 with p prime),primorial primes, Fermat primes and Mersenne primes,that is, prime numbers that are of the form 2p 1, wherep is an arbitrary prime. The LucasLehmer test is par-ticularly fast for numbers of this form. This is why thelargest known prime has almost always been a Mersenneprime since the dawn of electronic computers.Fermat primes are of the formFk = 22k+ 1,with k an arbitrary natural number. They are named afterPierre de Fermat who conjectured that all such numbersFk are prime. This was based on the evidence of the rstve numbers in this series3, 5, 17, 257, and 65,537being prime. However,F5 is composite and so are allother Fermat numbers that have been veried as of 2015.A regular n-gon is constructible using straightedge andcompass if and only if6 6 DISTRIBUTIONn = 2i mwhere m is a product of any number of distinct Fermatprimes and i is any natural number, including zero.The following table gives the largest known primes ofthe mentioned types. Some of these primes have beenfound using distributed computing. In 2009, the GreatInternet Mersenne Prime Search project was awarded aUS$100,000 prize for rst discovering a prime with atleast 10 million digits.[23] The Electronic Frontier Foun-dation also oers $150,000 and $250,000 for primeswithat least 100milliondigits and1billiondigits,respectively.[24] Some of the largest primes not known tohave any particular form (that is, no simple formula suchas that of Mersenne primes) have been found by taking apiece of semi-randombinary data, converting it to a num-ber n, multiplying it by 256k for some positive integerk, and searching for possible primes within the interval[256kn + 1, 256k(n + 1) 1].5.5 Integer factorizationMain article: Integer factorizationGiven a composite integer n, the task of providing one(or all) prime factors is referred to asfactorization ofn.Elliptic curve factorization is an algorithm relying onarithmetic on an elliptic curve.6 DistributionIn 1975, number theorist Don Zagier commented thatprimes bothgrow like weeds among the natural num-bers, seeming to obey no other law than thatof chance [but also] exhibit stunning regularity[and] that there are laws governing their behav-ior, and that they obey these laws with almostmilitary precision.[28]The distribution of primes in the large, such as the ques-tion how many primes are smaller than a given,largethreshold, is described by the prime number theorem, butno ecient formula for the n-th prime is known.There are arbitrarily long sequences of consecutive non-primes, as for every positive integer n the n consecutiveintegers from (n+1)! +2 to (n+1)! +n+1 (inclusive)are all composite (as (n + 1)! + k is divisible by k for kbetween 2 and n + 1 ).Dirichlets theorem on arithmetic progressions, in its ba-sic form, asserts that linear polynomialsp(n) = a + bnwith coprime integers a and b take innitely many primevalues. Stronger forms of the theorem state that the sumof the reciprocals of these prime values diverges, and thatdierent such polynomials with the same b have approx-imately the same proportions of primes.The corresponding question for quadratic polynomials isless well-understood.6.1 Formulas for primesMain article: formulas for primesThere is no known ecient formula for primes.For ex-ample, Mills theoremand a theoremof Wright assert thatthere are real constants A>1 and such thatA3n and2...22are prime for any natural number n. Here representsthe oor function, i.e., largest integer not greater than thenumber in question. The latter formula can be shownusing Bertrands postulate (proven rst by Chebyshev),which states that there always exists at least one primenumber p with n < p < 2n 2, for any natural number n >3. However, computing A or requires the knowledge ofinnitely many primes to begin with.[29] Another formulais based on Wilsons theorem and generates the number2 many times and all other primes exactly once.There is no non-constant polynomial, even in several vari-ables, that takes only prime values. However, there is a setof Diophantine equations in 9 variables and one parame-ter with the following property: the parameter is prime ifand only if the resulting system of equations has a solu-tion over the natural numbers. This can be used to obtaina single formula with the property that all its positive val-ues are prime.[30]6.2 Number of prime numbers belowagiven numberMain articles: Prime number theoremand Prime-counting functionThe prime counting function (n) is dened as the num-ber of primes not greater than n. For example (11) = 5,since there are ve primes less than or equal to 11. Thereare known algorithms to compute exact values of (n)faster than it would be possible to compute each primeup to n. The prime number theorem states that (n) isapproximately given by(n) nln n,in the sense that the ratio of (n) and the right hand frac-tion approaches 1 when n grows to innity. This implies6.4 Prime values of quadratic polynomials 720 000 40 000 60 000 80 000 1000002000400060008000A chart depicting (n) (blue), n / ln (n) (green) and Li(n) (red)that the likelihood that a number less than n is prime is(approximately) inversely proportional to the number ofdigits in n. A more accurate estimate for (n) is given bythe oset logarithmic integralLi(n) =n2dtln t.The prime number theorem also implies estimates for thesize of the n-th prime number pn (i.e., p1 = 2, p2 = 3,etc.): up to a bounded factor, pn grows like n log(n).[31]In particular, the prime gaps, i.e. the dierences pn pnof two consecutive primes, become arbitrarily large. Thislatter statement can also be seen in a more elementary wayby noting that the sequence n! + 2, n! + 3, , n! + n (forthe notation n! read factorial) consists of n 1 compositenumbers, for any natural number n.6.3 Arithmetic progressionsAn arithmetic progression is the set of natural numbersthat give the same remainder when divided by some xednumber q called modulus. For example,3, 12, 21, 30, 39, ...,is an arithmetic progression modulo q = 9. Except for 3,none of these numbers is prime, since 3 + 9n = 3(1 + 3n)so that the remaining numbers in this progression are allcomposite. (In general terms, all prime numbers above qare of the form q#n + m, where 0 < m < q#, and m hasno prime factor q.) Thus, the progressiona, a + q, a + 2q, a + 3q, can have innitely many primes only when a and q arecoprime, i.e., their greatest common divisor is one. Ifthis necessary condition is satised, Dirichlets theoremonarithmetic progressions asserts that the progression con-tains innitely many primes. The picture belowillustratesthis with q = 9: the numbers are wrapped around assoon as a multiple of 9 is passed. Primes are highlightedin red. The rows (=progressions) starting witha = 3,6, or 9 contain at most one prime number. In all otherrows (a = 1, 2, 4, 5, 7, and 8) there are innitely manyprime numbers. What is more, the primes are distributedequally among those rows in the long runthe density ofall primes congruent a modulo 9 is 1/6.Primenumbers (highlightedinred) inarithmeticprogressionmodulo 9.The GreenTao theorem shows that there are arbitrarilylong arithmetic progressions consisting of primes.[32] Anodd prime p is expressible as the sum of two squares, p= x2+ y2, exactly if p is congruent 1 modulo 4 (Fermatstheorem on sums of two squares).6.4 Prime values of quadratic polynomialsThe Ulam spiral. Red pixels show prime numbers. Primes of theform 4n2 2n + 41 are highlighted in blue.Euler noted that the functionn2+ n + 41gives prime numbers for 0 n < 40,[33][34] a fact lead-ing into deep algebraic number theory, more specicallyHeegner numbers. For bigger n, it does take compos-ite values. The Hardy-Littlewood conjecture F makes8 7 OPEN QUESTIONSanasymptoticpredictionabout thedensityofprimesamong the values of quadratic polynomials (with integercoecients a, b, and c)f(n) = ax2+ bx + cin terms of Li(n) and the coecients a, b, and c. How-ever, progress has proved hard to come by: no quadraticpolynomial (with a 0) is known to take innitely manyprime values. The Ulam spiral depicts all natural num-bers in a spiral-like way. Surprisingly, prime numberscluster on certain diagonals and not others, suggesting thatsome quadratic polynomials take prime values more oftenthan other ones.7 Open questions7.1 Zeta functionandthe Riemannhy-pothesisMain article: Riemann hypothesisThe Riemann zeta function (s) is dened as an innitePlot of the zeta function (s). At s=1, the function has a pole,that is to say, it tends to innity.sum(s) =n=11ns,where s is a complex number with real part bigger than1. It is a consequence of the fundamental theorem ofarithmetic that this sum agrees with the innite productpprime11 ps.The zeta function is closely related to prime numbers. Forexample, the aforementioned fact that there are innitelymany primes can also be seen using the zeta function: ifthere were only nitely many primes then (1) would havea nite value. However, the harmonic series 1 + 1/2 +1/3 + 1/4 + ... diverges (i.e., exceeds any given number),so there must be innitely many primes. Another exam-ple of the richness of the zeta function and a glimpse ofmodern algebraic number theory is the following identity(Basel problem), due to Euler,(2) =p11 p2=26.The reciprocal of (2), 6/2, is the probability that twonumbers selected at random are relatively prime.[35][36]The unprovenRiemannhypothesis, dating from 1859,states that except for s = 2, 4, ..., all zeroes of the -function have real part equal to 1/2. The connection toprime numbers is that it essentially says that the primesare as regularly distributed as possible. From a physi-cal viewpoint, it roughly states that the irregularity in thedistribution of primes only comes from random noise.From a mathematical viewpoint, it roughly states that theasymptotic distribution of primes (about x/log x of num-bers less than x are primes, the prime number theorem)also holds for much shorter intervals of length about thesquare root of x (for intervals near x). This hypothesis isgenerally believed to be correct. In particular, the sim-plest assumption is that primes should have no signicantirregularities without good reason.7.2 Other conjecturesFurther information: Category:Conjectures about primenumbersIn addition to the Riemann hypothesis, many more con-jectures revolving about primes have been posed. Oftenhaving an elementary formulation, many of these con-jectures have withstood a proof for decades:all four ofLandaus problems from 1912 are still unsolved. One ofthem is Goldbachs conjecture, which asserts that everyeven integern greater than 2 can be written as a sumof two primes. As of February 2011,this conjecturehas been veried for all numbers up to n = 2 1017.[37]Weaker statements than this have been proven, for ex-ample Vinogradovs theorem says that every sucientlylarge odd integer can be written as a sum of three primes.Chens theorem says that every suciently large evennumber can be expressed as the sum of a prime and asemiprime, the product of two primes. Also, any eveninteger can be written as the sum of six primes.[38] Thebranch of number theory studying such questions is calledadditive number theory.8.1 Arithmetic modulo a prime and nite elds 9Other conjectures deal with the question whether an inn-ity of prime numbers subject to certain constraints exists.It is conjectured that there are innitely many Fibonacciprimes[39] and innitely many Mersenne primes, but notFermat primes.[40] It is not known whether or not thereare an innite number of Wieferich primes and of primeEuclid numbers.A third type of conjectures concerns aspects of the distri-bution of primes. It is conjectured that there are innitelymany twin primes, pairs of primes with dierence 2 (twinprime conjecture). Polignacs conjecture is a strength-ening of that conjecture, it states that for every positiveinteger n, there are innitely many pairs of consecutiveprimes that dier by 2n.[41] It is conjectured there areinnitely many primes of the form n2+ 1.[42] These con-jectures are special cases of the broad Schinzels hypoth-esis H. Brocards conjecture says that there are alwaysat least four primes between the squares of consecutiveprimes greater than 2. Legendres conjecture states thatthere is a prime number between n2and (n + 1)2for everypositive integer n. It is implied by the stronger Cramrsconjecture.8 ApplicationsFor a long time, number theory in general, and the studyof prime numbers in particular, was seen as the canon-ical example of pure mathematics, with no applicationsoutside of the self-interest of studying the topic withthe exception of use of prime numbered gear teeth todistribute wear evenly. In particular, number theoristssuch as British mathematician G. H. Hardy prided them-selvesondoingworkthat hadabsolutelynomilitarysignicance.[43] However, this vision was shattered in the1970s, when it was publicly announced that prime num-bers could be used as the basis for the creation of publickey cryptography algorithms. Prime numbers are alsoused for hash tables and pseudorandom number gener-ators.Some rotor machines were designed with a dierent num-ber of pins on each rotor, with the number of pins on anyone rotor either prime, or coprime to the number of pinson any other rotor. This helped generate the full cycle ofpossible rotor positions before repeating any position.The International Standard Book Numbers work with acheck digit, which exploits the fact that 11 is a prime.8.1 Arithmetic modulo a prime and niteeldsMain article: Modular arithmeticModular arithmetic modies usual arithmetic by only us-ing the numbers{0, 1, 2, . . . , n 1},where n is a xed natural number called modulus. Cal-culating sums, dierences and products is done as usual,but whenever a negative number or a number greater thann 1 occurs, it gets replaced by the remainder after di-vision by n. For instance, for n = 7, the sum 3 + 5 is 1instead of 8, since 8 divided by 7 has remainder 1. Thisis referred to by saying 3 + 5 is congruent to 1 modulo7 and is denoted3 + 5 1 (mod7).Similarly, 6 + 1 0 (mod 7), 2 5 4 (mod 7), since3 + 7 = 4, and 3 4 5 (mod 7) as 12 has remain-der 5. Standard properties of addition and multiplicationfamiliar from the integers remain valid in modular arith-metic.In the parlance of abstract algebra, the above setof integers, which is also denotedZ/nZ, is therefore acommutative ring for any n. Division, however, is not ingeneral possible in this setting. For example, for n = 6,the equation3 x 2 (mod6),a solution x of which would be an analogue of 2/3, cannotbe solved, as one can see by calculating 3 0, ..., 3 5modulo 6. The distinctive feature of prime numbers isthe following: division is possible in modular arithmeticif and only if n is a prime. Equivalently, n is prime if andonly if all integers m satisfying 2 m n 1 are coprimeto n, i.e. their only common divisor is one. Indeed, for n= 7, the equation3 x 2 (mod7),has a unique solution, x = 3. Because of this, for anyprime p, Z/pZ (also denoted Fp) is called a eld or, morespecically, a nite eld since it contains nitely many,namely p, elements.Anumber of theorems can be derived frominspecting Fpin this abstract way. For example, Fermats little theorem,statingap1 1(modp)for any integer a not divisible by p, may be proved usingthese notions. This impliesp1a=1ap1 (p 1) 1 1 (modp).10 9 GENERALIZATIONSGiugas conjecture says that this equation is also a su-cient condition for p to be prime.Another consequenceof Fermats little theorem is the following: if p is a primenumber other than 2 and 5, 1/p is always a recurring deci-mal, whose period is p 1 or a divisor of p 1. The frac-tion1/p expressed likewise in base q (rather than base 10)has similar eect, provided that p is not a prime factor ofq. Wilsons theorem says that an integer p > 1 is primeif and only if the factorial (p 1)! + 1 is divisible by p.Moreover, an integer n > 4 is composite if and only if (n 1)! is divisible by n.8.2 Other mathematical occurrences ofprimesMany mathematical domains make great use of primenumbers. An example from the theory of nite groupsare the Sylow theorems:if G is a nite group and pnisthe highest power of the prime p that divides the order ofG, then G has a subgroup of order pn. Also, any group ofprime order is cyclic (Lagranges theorem).8.3 Public-key cryptographyMain article: Public key cryptographySeveral public-key cryptography algorithms, such as RSAand the DieHellman key exchange, are based on largeprimenumbers(forexample512bit primesarefre-quently used for RSA and 1024 bit primes are typical forDieHellman.). RSA relies on the assumption that itis much easier (i.e., more ecient) to perform the mul-tiplication of two (large) numbers x and y than to calcu-late x and y (assumed coprime) if only the product xy isknown. The DieHellman key exchange relies on thefact that there are ecient algorithms for modular expo-nentiation, while the reverse operation the discrete loga-rithm is thought to be a hard problem.8.4 Prime numbers in natureThe evolutionary strategy used by cicadas of the genusMagicicada make use of prime numbers.[44] These insectsspend most of their lives as grubs underground. Theyonly pupate and then emerge from their burrows after 7,13 or 17 years, at which point they y about, breed, andthen die after a few weeks at most.The logic for this isbelieved to be that the prime number intervals betweenemergences make it very dicult for predators to evolvethat could specialize as predators on Magicicadas.[45] IfMagicicadas appeared at a non-prime number intervals,say every 12 years, then predators appearing every 2, 3,4, 6, or 12 years would be sure to meet them. Over a200-year period, average predator populations during hy-pothetical outbreaks of 14- and 15-year cicadas would beup to 2% higher than during outbreaks of 13- and 17-year cicadas.[46] Though small, this advantage appears tohave been enough to drive natural selection in favour of aprime-numbered life-cycle for these insects.There is speculation that the zeros of the zeta functionare connected to the energy levels of complex quantumsystems.[47]9 GeneralizationsThe concept of prime number is so important that it hasbeen generalized in dierent ways in various branches ofmathematics. Generally, prime indicates minimality orindecomposability, in an appropriate sense. For exam-ple, the prime eld is the smallest subeld of a eld Fcontaining both 0 and 1.It is either Q or the nite eldwith p elements, whence the name.[48] Often a second,additional meaning is intended by using the word prime,namely that any object can be, essentially uniquely, de-composed into its prime components. For example, inknot theory, a prime knot is a knot that is indecomposablein the sense that it cannot be written as the knot sum oftwo nontrivial knots. Any knot can be uniquely expressedas a connected sum of prime knots.[49] Prime models andprime 3-manifolds are other examples of this type.9.1 Prime elements in ringsMain articles: Prime element and Irreducible elementPrime numbers give rise to two more general conceptsthat apply to elements of any commutative ringR, analgebraic structure where addition, subtraction and mul-tiplication are dened: prime elements and irreducible el-ements. An element p of R is called prime element if it isneither zero nor a unit (i.e., does not have a multiplicativeinverse) and satises the following requirement: given xand y in R such that p divides the product xy, then p di-vides x or y. An element is irreducible if it is not a unit andcannot be written as a product of two ring elements thatare not units.In the ring Z of integers, the set of primeelements equals the set of irreducible elements, which is{. . . , 11, 7, 5, 3, 2, 2, 3, 5, 7, 11, . . . } .In any ring R, any prime element is irreducible. The con-verse does not hold in general, but does hold for uniquefactorization domains.The fundamental theoremof arithmetic continues to holdin unique factorization domains. An example of such adomain is the Gaussian integers Z[i], that is, the set ofcomplex numbers of the forma +bi wherei denotesthe imaginary unit and a and b are arbitrary integers. Itsprime elements are known as Gaussian primes. Not every11prime (in Z) is a Gaussian prime: in the bigger ring Z[i],2 factors into the product of the two Gaussian primes (1 +i) and (1 i). Rational primes (i.e. prime elements in Z)of the form 4k + 3 are Gaussian primes, whereas rationalprimes of the form 4k + 1 are not.9.2 Prime idealsMain article: Prime idealsIn ring theory, the notion of number is generally replacedwith that of ideal. Prime ideals, which generalize primeelements in the sense that the principal ideal generated bya prime element is a prime ideal, are an important tool andobject of study in commutative algebra, algebraic numbertheory and algebraic geometry. The prime ideals of thering of integers are the ideals (0), (2), (3), (5), (7), (11), The fundamental theorem of arithmetic generalizes tothe LaskerNoether theorem, which expresses every idealin a Noetherian commutative ring as an intersection ofprimary ideals, which are the appropriate generalizationsof prime powers.[50]Prime ideals are the points of algebro-geometric objects,via the notion of the spectrumof a ring.[51] Arithmetic ge-ometry also benets from this notion, and many conceptsexist in both geometry and number theory. For example,factorization or ramication of prime ideals when liftedto an extension eld, a basic problem of algebraic num-ber theory, bears some resemblance with ramicationin geometry. Such ramication questions occur even innumber-theoretic questions solely concerned with inte-gers. For example, prime ideals in the ring of integers ofquadratic number elds can be used in proving quadraticreciprocity, a statement that concerns the solvability ofquadratic equationsx2 p ( modq),wherexis an integer andp andq are (usual) primenumbers.[52] Early attempts to prove Fermats Last Theo-rem climaxed when Kummer introduced regular primes,primes satisfying a certain requirement concerning thefailure of unique factorization in the ring consisting ofexpressionsa0 + a1 + + ap1p1,where a0, ..., ap1 are integers and is a complex numbersuch that p= 1.[53]9.3 ValuationsValuation theory studies certain functions from a eld Kto the real numbersR called valuations.[54] Every suchvaluation yields a topology on K, and two valuations arecalled equivalent if they yield the same topology. Aprimeof K (sometimes called a place of K) is an equivalenceclass of valuations. For example, the p-adic valuation ofa rational number q is dened to be the integer vp(q), suchthatq= pvp(q)rs,where both r and s are not divisible by p. For example,v3(18/7) = 2. The p-adic norm is dened as[nb 1]|q|p:= pvp(q).In particular, this norm gets smaller when a number ismultiplied by p, in sharp contrast to the usual absolutevalue(alsoreferredtoastheinniteprime). Whilecompleting Q (roughly, lling the gaps) with respect tothe absolute value yields the eld of real numbers, com-pleting with respect to the p-adic norm||p yields the eldof p-adic numbers.[55] These are essentially all possibleways to completeQ, by Ostrowskis theorem. Certainarithmetic questions related to Q or more general globalelds may be transferred back and forth to the completed(or local) elds. This local-global principle again under-lines the importance of primes to number theory.10 In the arts and literaturePrime numbers have inuenced many artists and writers.The French composer Olivier Messiaen used prime num-bers to create ametrical music through natural phenom-ena. In works such as La Nativit du Seigneur (1935) andQuatretudesderythme (194950), he simultaneouslyemploys motifs with lengths given by dierentprimenumbers to create unpredictable rhythms: the primes 41,43, 47 and 53 appear in the third tude, Neumes ryth-miques. According to Messiaen this way of composingwas inspired by the movements of nature, movements offree and unequal durations.[56]In his science ction novel Contact, NASA scientist CarlSagan suggested that prime numbers could be used asa means of communicating with aliens, an idea that hehad rst developed informally with American astronomerFrank Drake in 1975.[57] In the novel The Curious Inci-dent of the Dog in the Night-Time by Mark Haddon, thenarrator arranges the sections of the story by consecutiveprime numbers.[58]Many lms, such asCube, Sneakers, TheMirrorHasTwo Faces and ABeautiful Mind reect a popularfascinationwiththemysteriesofprimenumbersandcryptography.[59] Prime numbers are used as a metaphorfor loneliness and isolation in the Paolo Giordano novelThe Solitude of Prime Numbers, in which they are por-trayed as outsiders among integers.[60]12 12 NOTES11 See alsoAdlemanPomeranceRumely primality testBonses inequalityBrun sieveBurnside theoremChebotarevs density theoremChinese remainder theoremCullen numberIllegal primeList of prime numbersMersenne primeMultiplicative number theoryNumber eld sievePepins testPractical numberPrime k-tuplePrimon gasQuadratic residuosity problemRSA numberSmooth numberSuper-primeWoodall number12 Notes[1] Some sources also put |q|p:= evp(q). .[1] Dudley, Underwood (1978), Elementarynumbertheory(2nd ed.), W. H. Freeman and Co., ISBN 978-0-7167-0076-0, p. 10, section 2[2] Dudley 1978, Section 2, Theorem 2[3] Dudley 1978, Section 2, Lemma 5[4] See, for example, DavidE. Joyces commentary onEuclids Elements, Book VII, denitions 1 and 2.[5] https://primes.utm.edu/notes/faq/one.html[6] Weisstein, Eric W., Goldbach Conjecture, MathWorld[7] Riesel 1994, p. 36[8] Conway & Guy 1996, pp. 129130[9] Derbyshire, John (2003), The Prime Number Theorem,Prime Obsession: Bernhard Riemann and the Greatest Un-solved Problemin Mathematics, Washington, D.C.: JosephHenry Press, p. 33, ISBN 978-0-309-08549-6, OCLC249210614[10] https://primes.utm.edu/notes/faq/one.html[11] "Arguments for and against the primality of 1".[12] Why is the number one not prime?"[13] The LargestKnown Prime by Year: A Brief HistoryPrime Curios!: 1701405727 (39-digits)[14] Forinstance, Beilerwritesthat numbertheorist ErnstKummer loved his ideal numbers, closely related to theprimes, because they had not soiled themselves with anypractical applications, and Katz writes that Edmund Lan-dau, known for his work on the distribution of primes,loathed practical applications of mathematics, and forthis reason avoided subjects such as geometry that hadalready shown themselves to be useful. Beiler,AlbertH. (1966), RecreationsintheTheoryofNumbers: TheQueenofMathematicsEntertains, Dover, p. 2, ISBN9780486210964. Katz, Shaul (2004), Berlin rootsZionist incarnation: the ethos of pure mathematics and thebeginnings of the Einstein Institute of Mathematics at theHebrew University of Jerusalem, Science in Context 17(1-2): 199234, doi:10.1017/S0269889704000092, MR2089305.[15] Letter in Latin from Goldbach to Euler, July 1730.[16] Furstenberg 1955[17] Ribenboim 2004, p. 4[18] James Williamson (translator and commentator), The Ele-ments of Euclid, With Dissertations, Clarendon Press, Ox-ford, 1782, page 63, English translation of Euclids proof[19] Hardy, Michael;Woodgold, Catherine (2009). PrimeSimplicity. Mathematical Intelligencer31 (4): 4452.doi:10.1007/s00283-009-9064-8.[20] Apostol, TomM. (1976), Introduction to Analytic NumberTheory, Berlin, New York:Springer-Verlag, ISBN 978-0-387-90163-3, Section 1.6, Theorem 1.13[21] Apostol 1976, Section 4.8, Theorem 4.12[22] (Lehmer 1909).[23] Record 12-Million-Digit Prime Number Nets $100,000Prize. ElectronicFrontierFoundation. October14,2009. Retrieved 2010-01-04.[24] EFF Cooperative Computing Awards. Electronic Fron-tier Foundation. Retrieved 2010-01-04.[25] Chris K. Caldwell. The Top Twenty: Factorial.Primes.utm.edu. Retrieved 2013-02-05.[26] Chris K. Caldwell. The TopTwenty: Primorial.Primes.utm.edu. Retrieved 2013-02-05.[27] Chris K. Caldwell. The Top Twenty: Twin Primes.Primes.utm.edu. Retrieved 2013-02-05.13[28] Havil 2003, p. 171[29] https://books.google.com/books?id=oLKlk5o6WroC&pg=PA13#v=onepage&q&f=false p. 15[30] Matiyasevich, Yuri V. (1999), Formulas for Prime Num-bers, in Tabachnikov, Serge, Kvant Selecta: Algebra andAnalysis, II, American Mathematical Society, pp. 1324,ISBN 978-0-8218-1915-9.[31] (Tom M. Apostol 1976), Section 4.6, Theorem 4.7[32] (Ben Green & Terence Tao 2008).[33] Hua (2009), pp. 176177"[34] See list of values, calculated by Wolfram Alpha[35] Caldwell, Chris. What is the probability thatgcd(n,m)=1?". The Prime Pages. Retrieved2013-09-06.[36] C. S. Ogilvy & J. T. Anderson Excursions in Number The-ory, pp. 2935, Dover Publications Inc., 1988 ISBN 0-486-25778-9[37] Toms Oliveira e Silva (2011-04-09). Goldbach conjec-ture verication. Ieeta.pt. Retrieved 2011-05-21.[38] Ramar, O. (1995), On nirel'mans constant, Annalidella Scuola Normale Superiore di Pisa. Classe di Scienze.Serie IV 22 (4): 645706, retrieved 2008-08-22.[39] Caldwell, Chris, The Top Twenty: Lucas Number at ThePrime Pages.[40] E.g., see Guy 1981, problem A3, pp. 78[41] Tattersall, J.J. (2005), Elementary number theory in ninechapters, Cambridge University Press, ISBN 978-0-521-85014-8, p. 112[42] Weisstein, Eric W., Landaus Problems, MathWorld.[43] Hardy 1940 No one has yet discovered any warlike pur-pose to be served by the theory of numbers or relativ-ity, and it seems unlikely that anyone will do so for manyyears.[44] Goles, E.; Schulz, O.; Markus, M. (2001). Prime numberselection of cycles in a predator-prey model. Complexity6 (4): 3338. doi:10.1002/cplx.1040.[45] Paulo R. A. Campos, Viviane M. de Oliveira, RonaldoGiro, andDouglas S. Galvo. (2004), Emergenceof Prime Numbers as the Result of EvolutionaryStrategy, Physical ReviewLetters 93 (9): 098107,arXiv:q-bio/0406017, Bibcode:2004PhRvL..93i8107C,doi:10.1103/PhysRevLett.93.098107.[46] Invasion of the Brood. The Economist. May 6, 2004.Retrieved 2006-11-26.[47] Ivars Peterson (June 28, 1999). The Return of Zeta.MAA Online. Retrieved 2008-03-14.[48] Lang, Serge (2002), Algebra, Graduate Texts in Math-ematics 211, Berlin, New York: Springer-Verlag, ISBN978-0-387-95385-4, MR 1878556, Section II.1, p. 90[49] Schubert, H. Die eindeutige Zerlegbarkeit eines Knotensin Primknoten. S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57104.[50] Eisenbud 1995, section 3.3.[51] Shafarevich, Basic Algebraic Geometry volume 2(Schemes and Complex Manifolds), p. 5, section V.1[52] Neukirch, Algebraic Number theory, p. 50, Section I.8[53] Neukirch, Algebraic Number theory, p. 38, Section I.7[54] Endler, Valuation Theory, p. 1[55] Gouvea:p-adic numbers:an introduction, Chapter 3, p.43[56] Hill, ed. 1995[57] Carl Pomerance, PrimeNumbersandtheSearchforExtraterrestrial Intelligence, Retrieved on December 22,2007[58] Mark Sarvas, Book Review: The Curious Incident of theDog in the Night-Time, at The Modern Word, Retrievedon March 30, 2012[59] The music of primes,Marcus du Sautoy's selection oflms featuring prime numbers.[60] Introducing Paolo Giordano. Books Quarterly.13 ReferencesApostol, Thomas M. (1976), Introduction to Ana-lytic Number Theory, New York: Springer, ISBN 0-387-90163-9Conway, John Horton; Guy, Richard K. (1996), TheBookofNumbers, New York: Copernicus, ISBN978-0-387-97993-9Crandall, Richard; Pomerance, Carl (2005), PrimeNumbers: AComputational Perspective (2nd ed.),Berlin, New York: Springer-Verlag, ISBN 978-0-387-25282-7Derbyshire, John (2003), Primeobsession, JosephHenry Press,Washington,DC, ISBN 978-0-309-08549-6, MR 1968857Eisenbud, David (1995), Commutative algebra,Graduate Texts in Mathematics150, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94268-1,MR 1322960Fraleigh, John B. (1976), A First Course In AbstractAlgebra (2nd ed.), Reading: Addison-Wesley, ISBN0-201-01984-1Furstenberg, Harry (1955), On the innitudeof primes, TheAmericanMathematical Monthly(Mathematical Associationof America) 62(5):353, doi:10.2307/2307043, JSTOR 230704314 14 EXTERNAL LINKSGreen, Ben; Tao, Terence (2008), Theprimes contain arbitrarily long arithmeticprogressions, Annals of Mathematics 167(2): 481547, arXiv:math.NT/0404188,doi:10.4007/annals.2008.167.481Gowers, Timothy(2002), Mathematics: AVeryShort Introduction, Oxford University Press, ISBN978-0-19-285361-5Guy, RichardK. (1981), UnsolvedProblems inNumber Theory, Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-90593-8Havil, Julian(2003), Gamma: ExploringEulersConstant, Princeton University Press, ISBN 978-0-691-09983-5Hardy, Godfrey Harold (1908), A Course of PureMathematics, CambridgeUniversityPress, ISBN978-0-521-09227-2Hardy, Godfrey Harold (1940), A MathematiciansApology, Cambridge University Press, ISBN 978-0-521-42706-7Herstein, I. N. (1964), Topics In Algebra, Waltham:Blaisdell Publishing Company, ISBN 978-1114541016Hill, Peter Jensen, ed. (1995), The Messiaen com-panion, Portland, Or: Amadeus Press, ISBN 978-0-931340-95-6Hua, L. K. (2009), Additive Theory of Prime Num-bers, Translations of Mathematical Monographs 13,AMS Bookstore, ISBN 978-0-8218-4942-2Lehmer, D. H. (1909), Factor table for the rst tenmillions containing the smallest factor of every num-ber not divisible by 2, 3, 5, or 7 between the limits 0and 10017000, Washington, D.C.: Carnegie Insti-tution of WashingtonMcCoy, Neal H. (1968), Introduction To Modern Al-gebra, Revised Edition, Boston: Allyn and Bacon,LCCN 68-15225Narkiewicz, Wladyslaw (2000), The development ofprimenumbertheory: fromEuclidtoHardyandLittlewood, Springer Monographs in Mathematics,Berlin, New York: Springer-Verlag, ISBN 978-3-540-66289-1Ribenboim, Paulo (2004), The little book of biggerprimes, Berlin, New York: Springer-Verlag, ISBN978-0-387-20169-6Riesel, Hans (1994), Prime numbers and com-puter methods for factorization, Basel, Switzerland:Birkhuser, ISBN 978-0-8176-3743-9Sabbagh, Karl (2003), The Riemann hypothesis, Far-rar,Straus and Giroux,New York,ISBN 978-0-374-25007-2, MR 1979664du Sautoy, Marcus (2003), The music of the primes,HarperCollins Publishers, ISBN 978-0-06-621070-4, MR 206013414 External linksHazewinkel, Michiel, ed. (2001), Prime number,Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Caldwell, Chris, The Prime Pages atprimes.utm.edu.Prime Numbers on In Our Time at the BBC.An Introduction to Analytic Number Theory, by IlanVardi and Cyril BanderierPlus teacher and student package: prime numbersfrom Plus,the free online mathematics magazineproduced by the MillenniumMathematics Project atthe University of Cambridge.14.1 Prime number generators and calcu-latorsPrime Number Checker identies the smallest primefactor of a number.Fast Online primality test with factorization makesuse of the Elliptic Curve Method (up to thousand-digits numbers, requires Java).Huge database of prime numbersPrime Numbers up to 1 trillion1515 Text and image sources, contributors, and licenses15.1 Text Prime number Source: https://en.wikipedia.org/wiki/Prime_number?oldid=675838557Contributors: DamianYerrick, AxelBoldt,LC~enwiki, Lee Daniel Crocker, Mav, Bryan Derksen, Zundark, Tarquin, Hajhouse, XJaM, Arvindn, Christian List, The Ostrich, BoleslavBobcik, FvdP, Imran, Karl Palmen, Olivier, Paul Ebermann, Jim McKeeth, Bdesham, Patrick, Michael Hardy, Pwlfong, Wshun, Dougmer-ritt, Dominus, Ixfd64, Dcljr, TakuyaMurata, GTBacchus, Eric119, Tengai~enwiki, Pcb21, Egil, Ahoerstemeier, Stevenj, Docu, Jpatokal,Samuelsen, Snoyes, 5ko, Angela, Kingturtle, Lupinoid, Nikai, Rotem Dan, Cimon Avaro, Jacquerie27, Schneelocke, Jengod, Revolver,Charles Matthews, Timwi, Dcoetzee, Dysprosia, Malcohol, Fuzheado, Evgeni Sergeev, Tpbradbury, Furrykef, Taxman, VeryVerily, Fi-bonacci, Paul-L~enwiki, Darkhorse, Sabbut, Bevo, Shizhao, Warofdreams, Optim, Pakaran, Johnleemk, AnthonyQBachler, Garo, Jeq,Gromlakh, Robbot, Sander123, Chrism, Fredrik, Chris 73, Schutz, Chocolateboy, Calmypal, Lowellian, Gandalf61, Georg Muntingh,Merovingian, Henrygb, Bkell, Intangir, Mark Krueger, Saforrest, JackofOz, Kent Wang, PrimeFan, Seth Ilys, Tobias Bergemann, Giftlite,Dbenbenn, JamesMLane, Abigail-II, Reub2000, Hagedis, MSGJ, Herbee, 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