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Prime number

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Number divisible only by 1 or itself

"Prime" redirects here. For other uses, seePrime (disambiguation).

Groups of two to twelve dots, showing that the composite numbers of dots (4, 6, 8, 9, 10, and 12) can be arranged into rectangles but prime numbers cannot — Composite numbers can be arranged intorectangles but prime numbers cannot.

Aprime number (or aprime) is anatural number greater than 1 that is not aproduct of two smaller natural numbers. A natural number greater than 1 that is not prime is called acomposite number. For example, 5 is prime because the only ways of writing it as a product,1 × 5 or5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central innumber theory because of thefundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can befactorized as a product of primes that is uniqueup to their order.

The property of being prime is calledprimality. A simple but slowmethod of checking the primality of a given number⁠ $n {\displaystyle n}$ ⁠, calledtrial division, tests whether⁠ $n {\displaystyle n}$ ⁠ is a multiple of any integer between 2 and⁠ ${\sqrt {n}}$ ⁠. Faster algorithms include theMiller–Rabin primality test, which is fast but has a small chance of error, and theAKS primality test, which always produces the correct answer inpolynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such asMersenne numbers. As of October 2024^[update] thelargest known prime number is a Mersenne prime with 41,024,320decimal digits.^[1]^[2]

There areinfinitely many primes, asdemonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is theprime number theorem, proven at the end of the 19th century, which says roughly that theprobability of a randomly chosen large number being prime is inverselyproportional to its number of digits, that is, to itslogarithm.

Several historical questions regarding prime numbers are still unsolved. These includeGoldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and thetwin prime conjecture, that there are infinitely many pairs of primes that differ by two. Such questions spurred the development of various branches of number theory, focusing onanalytic oralgebraic aspects of numbers. Primes are used in several routines ininformation technology, such aspublic-key cryptography, which relies on the difficulty offactoring large numbers into their prime factors. Inabstract algebra, objects that behave in a generalized way like prime numbers includeprime elements andprime ideals.

Definition and examples

Main article:List of prime numbers

Anatural number (1, 2, 3, 4, 5, 6, etc.) is called aprime number (or aprime) if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are calledcomposite numbers.^[3] 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,^[4] 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.^[5] For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,^[6] 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 and6 = 2 × 3 are both composite.

refer to caption — Demonstration, withCuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly

Thedivisors of a natural number⁠ $n {\displaystyle n}$ ⁠ are the natural 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 leads to an equivalent definition of prime numbers: they are the numbers with exactly two positivedivisors. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition.^[7] Yet another way to express the same thing is that a number⁠ $n {\displaystyle n}$ ⁠ is prime if it is greater than one and if none of the numbers $2,3,\dots ,n-1$ divides⁠ $n {\displaystyle n}$ ⁠ evenly.^[8]

The first 25 prime numbers (all the prime numbers less than 100) are:^[9]

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 (sequenceA000040 in theOEIS).

Noeven number⁠ $n {\displaystyle n}$ ⁠ greater than 2 is prime because any such number can be expressed as the product⁠ $2\times n/2$ ⁠. Therefore, every prime number other than 2 is anodd number, and is called anodd prime.^[10] Similarly, when written in the usualdecimal system, all prime numbers larger than 5 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.^[11]

Theset of all primes is sometimes denoted by $\mathbf {P}$ (aboldface capital P)^[12] or by $\mathbb {P}$ (ablackboard bold capital P).^[13]

History

TheRhind Mathematical Papyrus, from around 1550 BC, hasEgyptian fraction expansions of different forms for prime and composite numbers.^[14] However, the earliest surviving records of the study of prime numbers come from theancient Greek mathematicians, who called themprōtos arithmòs (πρῶτος ἀριθμὸς).Euclid'sElements (c. 300 BC) proves theinfinitude of primes and thefundamental theorem of arithmetic, and shows how to construct aperfect number from a Mersenne prime.^[15] Another Greek invention, theSieve of Eratosthenes, is still used to construct lists ofprimes.^[16]^[17]

Around 1000 AD, theIslamic mathematicianIbn al-Haytham (Alhazen) foundWilson's theorem, characterizing the prime numbers as the numbers⁠ $n {\displaystyle n}$ ⁠ that evenly divide⁠ $(n-1)!+1$ ⁠. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.^[18] Another Islamic mathematician,Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit.^[17]Fibonacci took the innovations from Islamic mathematics to Europe. His bookLiber Abaci (1202) was the first to describetrial division for testing primality, again using divisors only up to the square root.^[17]

In 1640Pierre de Fermat stated (without proof)Fermat's little theorem (later proved byLeibniz andEuler).^[19] Fermat also investigated the primality of theFermat numbers⁠ $2^{2^{n}}+1$ ⁠,^[20] andMarin Mersenne studied the Mersenne primes, prime numbers of the form $2^{p}-1$ with⁠ $p {\displaystyle p}$ ⁠ itself a prime.^[21]Christian Goldbach formulatedGoldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.^[22] Euler proved Alhazen's conjecture (now theEuclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes.^[15] He introduced methods frommathematical analysis to this area in his proofs of the infinitude of the primes and thedivergence of the sum of the reciprocals of the primes⁠ ${\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+\cdots$ ⁠.^[23] 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 $x {\displaystyle x}$ ⁠ isasymptotic to⁠ $x/\log x$ ⁠, where $\log x$ is thenatural logarithm of⁠ $x {\displaystyle x}$ ⁠. A weaker consequence of this high density of primes wasBertrand's postulate, that for every $n>1$ there is a prime between⁠ $n {\displaystyle n}$ ⁠ and⁠ $2n$ ⁠, proved in 1852 byPafnuty Chebyshev.^[24] Ideas ofBernhard Riemann in his1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely relatedRiemann hypothesis remains unproven, Riemann's outline was completed in 1896 byHadamard andde la Vallée Poussin, and the result is now known as theprime number theorem.^[25] Another important 19th century result wasDirichlet's theorem on arithmetic progressions, that certainarithmetic progressions contain infinitely many primes.^[26]

Many mathematicians have worked onprimality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms includePépin's test for Fermat numbers (1877),^[27]Proth's theorem (c. 1878),^[28] theLucas–Lehmer primality test (originated 1856), and the generalizedLucas primality test.^[17]

Since 1951 all thelargest known primes have been found using these tests oncomputers.^[a] The search for ever larger primes has generated interest outside mathematical circles, through theGreat Internet Mersenne Prime Search and otherdistributed computing projects.^[9]^[30] The idea that prime numbers had few applications outside ofpure mathematics^[b] was shattered in the 1970s whenpublic-key cryptography and theRSA cryptosystem were invented, using prime numbers as their basis.^[33]

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.^[16]^[34]^[35] The mathematical theory of prime numbers also moved forward with theGreen–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, andYitang Zhang's 2013 proof that there exist infinitely manyprime gaps of bounded size.^[36]

Primality of one

Most early Greeks did not even consider 1 to be a number,^[37]^[38] so they could not consider its primality. A few scholars in the Greek and later Roman tradition, includingNicomachus,Iamblichus,Boethius, andCassiodorus, also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider⁠ $2 {\displaystyle 2}$ ⁠ to be prime either. However, Euclid and a majority of the other Greek mathematicians considered⁠ $2 {\displaystyle 2}$ ⁠ as prime. Themedieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.^[37] By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and by the 17th century some of them included it as the first prime number.^[39] In the mid-18th century,Christian Goldbach listed 1 as prime in his correspondence withLeonhard Euler;^[40] however, Euler himself did not consider 1 to be prime.^[41] Many 19th century mathematicians still considered 1 to be prime,^[42] andDerrick Norman Lehmer included 1 in hislist of primes less than ten million published in 1914.^[43] Lists of primes that included 1 continued to be published as recentlyas 1956.^[44]^[45] However, 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".^[42]

If 1 were to be considered a prime, many statements involving primes would need to be awkwardly reworded. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with any number of copies of 1.^[42] Similarly, thesieve 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.^[45] Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas forEuler's totient function or for thesum of divisors function are different for prime numbers than they are for 1.^[46]

Elementary properties

Unique factorization

Main article:Fundamental theorem of arithmetic

Writing a number as a product of prime numbers is called aprime factorization of the number.^[47] For example:

{\begin{aligned}50&=2\times 5\times 5\\&=2\times 5^{2}.\end{aligned}}

The terms in the product are calledprime factors. The same prime factor may occur more than once; this example has two copies of the prime factor $5. {\displaystyle 5.}$ When a prime occurs multiple times,exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, $5^{2}$ denotes thesquare or second power of⁠ $5 {\displaystyle 5}$ ⁠.^[47]

The central importance of prime numbers to number theory and mathematics in general stems from thefundamental theorem of arithmetic.^[48] 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.^[49] So, although there are many different ways of finding a factorization using aninteger factorization algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers.^[50]

Some proofs of the uniqueness of prime factorizations are based onEuclid'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).^[51] 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.^[52]

Infinitude

Main article:Euclid's theorem

There areinfinitely 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 asEuclid's theorem in honor of the ancient Greek mathematicianEuclid, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including ananalytical proof byEuler,Goldbach's proof based onFermat numbers,^[53]Furstenberg's proof using general topology,^[54] andKummer's elegant proof.^[55]

Euclid's proof^[56] shows that everyfinite 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 $p_{1},p_{2},\ldots ,p_{n},$ this gives the number

N=1+p_{1}\cdot p_{2}\cdots p_{n}.

By the fundamental theorem,⁠ $N {\displaystyle N}$ ⁠ has a prime factorization

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 calledEuclid numbers.^[57] The first five of them are prime, but the sixth,

1+{\big (}2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13{\big )}=30031=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-constantpolynomial, even in several variables, that takesonly prime values.^[58] However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based onWilson's theorem and generates the number 2 many times and all other primes exactly once.^[59] There is also a set ofDiophantine 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 itspositive values are prime.^[58]

Other examples of prime-generating formulas come fromMills' theorem and a theorem ofWright. These assert that there are real constants $A>1$ and $\mu$ such that

\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.^[60] Here $\lfloor {}\cdot {}\rfloor$ represents thefloor 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 $\mu .$ ^[58]

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 ofLandau's problems from 1912 are still unsolved.^[61] One of them isGoldbach's conjecture, which asserts that every even integer⁠ $n {\displaystyle n}$ ⁠ greater than⁠ $2 {\displaystyle 2}$ ⁠ can be written as a sum of two primes.^[62] As of 2014^[update], this conjecture has been verified for all numbers up to $n=4\cdot 10^{18}.$ ^[63] 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.^[64]Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and asemiprime (the product of two primes).^[65] Also, any even integer greater than 10 can be written as the sum of six primes.^[66] The branch of number theory studying such questions is calledadditive number theory.^[67]

Another type of problem concernsprime gaps, the differences between consecutive primes.The existence of arbitrarily large prime gaps can be seen by noting that the sequence $n!+2,n!+3,\dots ,n!+n$ consists of $n-1$ composite numbers, for any natural number $n . {\displaystyle n.}$ ^[68] However, large prime gaps occur much earlier than this argument shows.^[69] For example, the first prime gap of length 8 is between the primes 89 and 97,^[70] much smaller than $8!=40320.$ It is conjectured that there are infinitely manytwin primes, pairs of primes with difference 2; this is thetwin 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 $2k.$ ^[71]Andrica's conjecture,^[71]Brocard's conjecture,^[72]Legendre's conjecture,^[73] andOppermann's conjecture^[72] all suggest that the largest gaps between primes from 1 to⁠ $n {\displaystyle n}$ ⁠ should be at most approximately ${\sqrt {n}},$ a result that is known to follow from the Riemann hypothesis, while the much strongerCramér conjecture sets the largest gap size at⁠ $O((\log n)^{2})$ ⁠.^[71] Prime gaps can be generalized toprime⁠ $k {\displaystyle k}$ ⁠-tuples, patterns in the differences among more than two prime numbers. Their infinitude and density are the subject of thefirst Hardy–Littlewood conjecture, which can be motivated by theheuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.^[74]

Analytic properties

Analytic number theory studies number theory through the lens ofcontinuous functions,limits,infinite series, and the related mathematics of the infinite andinfinitesimal.

This area of study began withLeonhard Euler and his first major result, the solution to theBasel problem.The problem asked for the value of the infinite sum $1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots ,$ which today can be recognized as the value $\zeta (2)$ of theRiemann zeta function. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, theRiemann hypothesis. Euler showed that⁠ $\zeta (2)=\pi ^{2}/6$ ⁠.^[75]The reciprocal of this number,⁠ $6/\pi ^{2}$ ⁠, is the limiting probability that two random numbers selected uniformly from a large range arerelatively prime (have no factors in common).^[76]

The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by theprime number theorem, but no efficientformula for the⁠ $n {\displaystyle n}$ ⁠-th prime is known.Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials

p(n)=a+bn

with relatively prime integers⁠ $a {\displaystyle a}$ ⁠ and⁠ $b {\displaystyle b}$ ⁠ take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same⁠ $b {\displaystyle b}$ ⁠ have approximately the same proportions of primes.Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often.

Analytical proof of Euclid's theorem

Euler's proof that there are infinitely many primes considers the sums ofreciprocals of primes,

{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+\cdots +{\frac {1}{p}}.

Euler showed that, for any arbitraryreal number⁠ $x {\displaystyle x}$ ⁠, there exists a prime⁠ $p {\displaystyle p}$ ⁠ for which this sum is greater than⁠ $x {\displaystyle x}$ ⁠.^[77] This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every ⁠ $x {\displaystyle x}$ ⁠.The growth rate of this sum is described more precisely byMertens' second theorem.^[78] For comparison, the sum

{\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}

does not grow to infinity as⁠ $n {\displaystyle n}$ ⁠ goes to infinity (see theBasel problem). In this sense, prime numbers occur more often than squares of natural numbers,although both sets are infinite.^[79]Brun's theorem states that the sum of the reciprocals oftwin primes,

\left({{\frac {1}{3}}+{\frac {1}{5}}}\right)+\left({{\frac {1}{5}}+{\frac {1}{7}}}\right)+\left({{\frac {1}{11}}+{\frac {1}{13}}}\right)+\cdots ,

is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve thetwin prime conjecture, that there exist infinitely many twin primes.^[79]

Number of primes below a given bound

Main articles:Prime number theorem andPrime-counting function

Therelative error of ${\tfrac {n}{\log n}}$ and the logarithmic integral $\operatorname {Li} (n)$ as approximations to theprime-counting function. Both relative errors decrease to zero as⁠ $n {\displaystyle n}$ ⁠ grows, but the convergence to zero is much more rapid for the logarithmic integral.

{\displaystyle {\tfrac {n}{\log n}}} — Therelative error of ${\tfrac {n}{\log n}}$ and the logarithmic integral $\operatorname {Li} (n)$ as approximations to theprime-counting function. Both relative errors decrease to zero as⁠ $n {\displaystyle n}$ ⁠ grows, but the convergence to zero is much more rapid for the logarithmic integral.

Theprime-counting function $\pi (n)$ is defined as the number of primes not greater than⁠ $n {\displaystyle n}$ ⁠.^[80] For example,⁠ $\pi (11)=5$ ⁠, since there are five primes less than or equal to 11. Methods such as theMeissel–Lehmer algorithm can compute exact values of $\pi (n)$ faster than it would be possible to list each prime up to⁠ $n {\displaystyle n}$ ⁠.^[81] Theprime number theorem states that $\pi (n)$ is asymptotic to⁠ $n/\log n$ ⁠, which is denoted as

\pi (n)\sim {\frac {n}{\log n}},

and means that the ratio of $\pi (n)$ to the right-hand fractionapproaches 1 as⁠ $n {\displaystyle n}$ ⁠ grows to infinity.^[82] This implies that the likelihood that a randomly chosen number less than⁠ $n {\displaystyle n}$ ⁠ is prime is (approximately) inversely proportional to the number of digits in ⁠ $n {\displaystyle n}$ ⁠.^[83]It also implies that the⁠ $n {\displaystyle n}$ ⁠th prime number is proportional to $n\log n$ ^[84]and therefore that the average size of a prime gap is proportional to⁠ $\log n$ ⁠.^[69]A more accurate estimate for $\pi (n)$ is given by theoffset logarithmic integral^[82]

\pi (n)\sim \operatorname {Li} (n)=\int _{2}^{n}{\frac {dt}{\log t}}.

Arithmetic progressions

Main articles:Dirichlet's theorem on arithmetic progressions andGreen–Tao theorem

Anarithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.^[85] This difference is called themodulus of the progression.^[86] For example,

3,12,21,30,39,...,

is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression

a,a+q,a+2q,a+3q,\dots

can have more than one prime only when its remainder⁠ $a {\displaystyle a}$ ⁠ and modulus⁠ $q {\displaystyle q}$ ⁠ are relatively prime. If they are relatively prime,Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.^[87]

Prime numbers in arithmetic progression mod 9

Primes in the arithmetic progressions modulo 9. Each row of the thin horizontal band shows one of the nine possible progressions mod 9, with prime numbers marked in red. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.

TheGreen–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.^[36]^[88]

Prime values of quadratic polynomials

The Ulam spiral — TheUlam spiral. Prime numbers (orange) cluster on some diagonals and not others. Prime values of $4n^{2}-2n+41$ are shown in blue.

{\displaystyle 4n^{2}-2n+41} — TheUlam spiral. Prime numbers (orange) cluster on some diagonals and not others. Prime values of $4n^{2}-2n+41$ are shown in blue.

Euler noted that the function

n^{2}-n+41

yields prime numbers for⁠ $1\leq n\leq 40$ ⁠, although composite numbers appear among its later values.^[89]^[90] The search for an explanation for this phenomenon led to the deepalgebraic number theory ofHeegner numbers and theclass number problem.^[91] TheHardy–Littlewood conjecture F predicts the density of primes among the values ofquadratic polynomials with integercoefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.^[92]

TheUlam spiral^[93] arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.^[92]

Zeta function and the Riemann hypothesis

Main article:Riemann hypothesis

Plot of the absolute values of the zeta function, showing some of its features

One of the most famous unsolved questions in mathematics, dating from 1859, and one of theMillennium Prize Problems, is theRiemann hypothesis, which asks where thezeros of theRiemann zeta function $\zeta (s)$ are located.This function is ananalytic function on thecomplex numbers. For complex numbers⁠ $s {\displaystyle s}$ ⁠ with real part greater than one it equals both aninfinite sum over all integers, and aninfinite product over the prime numbers,

\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}.

This equality between a sum and a product, discovered by Euler, is called anEuler product.^[94] The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers.^[95]It leads to another proof that there are infinitely many primes: if there were only finitely many,then the sum-product equality would also be valid at⁠ $s=1$ ⁠, but the sum would diverge (it is theharmonic series⁠ $1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\dots$ ⁠) while the product would be finite, a contradiction.^[96]

The Riemann hypothesis states that thezeros of the zeta-function are all either negative even numbers, or complex numbers withreal part equal to 1/2.^[97] The original proof of theprime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1,^[98]^[99] although other more elementary proofs have been found.^[100] The prime-counting function can be expressed byRiemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term.^[101] In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and theasymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of⁠ $x {\displaystyle x}$ ⁠ for intervals near a number⁠ $x {\displaystyle x}$ ⁠).^[99]

Abstract algebra

Modular arithmetic and finite fields

Main article:Modular arithmetic

Modular arithmetic modifies usual arithmetic by only using the numbers⁠ $\{0,1,2,\dots ,n-1\}$ ⁠, for a natural number⁠ $n {\displaystyle n}$ ⁠ called the modulus.Any other natural number can be mapped into this system by replacing it by its remainder after division by⁠ $n {\displaystyle n}$ ⁠.^[102] Modular sums, differences and products are calculated by performing the same replacement by the remainder on the result of the usual sum, difference, or product of integers.^[103] Equality of integers corresponds tocongruence in modular arithmetic:⁠ $x {\displaystyle x}$ ⁠ and⁠ $y {\displaystyle y}$ ⁠ are congruent (written $x\equiv y$ mod⁠ $n {\displaystyle n}$ ⁠) when they have the same remainder after division by⁠ $n {\displaystyle n}$ ⁠.^[104] However, in this system of numbers,division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number 7 as modulus, division by 3 is possible:⁠ $2/3\equiv 3{\bmod {7}}$ ⁠, becauseclearing denominators by multiplying both sides by 3 gives the valid formula⁠ $2\equiv 9{\bmod {7}}$ ⁠. However, with the composite modulus 6, division by 3 is impossible. There is no valid solution to $2/3\equiv x{\bmod {6}}$ : clearing denominators by multiplying by 3 causes the left-hand side to become 2 while the right-hand side becomes either 0 or 3. In the terminology ofabstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms afield or, more specifically, afinite field, while other moduli only give aring but not a field.^[105]

Several theorems about primes can be formulated using modular arithmetic. For instance,Fermat's little theorem states that if $a\not \equiv 0$ (mod⁠ $p {\displaystyle p}$ ⁠), then $a^{p-1}\equiv 1$ (mod⁠ $p {\displaystyle p}$ ⁠).^[106] Summing this over all choices of⁠ $a {\displaystyle a}$ ⁠ gives the equation

\sum _{a=1}^{p-1}a^{p-1}\equiv (p-1)\cdot 1\equiv -1{\pmod {p}},

valid whenever⁠ $p {\displaystyle p}$ ⁠ is prime.Giuga's conjecture says that this equation is also a sufficient condition for⁠ $p {\displaystyle p}$ ⁠ to be prime.^[107]Wilson's theorem says that an integer $p>1$ is prime if and only if thefactorial $(p-1)!$ is congruent to $-1$ mod⁠ $p {\displaystyle p}$ ⁠. For a composite number ⁠ $n=r\cdot s$ ⁠ this cannot hold, since one of its factors divides bothn and⁠ $(n-1)!$ ⁠, and so $(n-1)!\equiv -1{\pmod {n}}$ is impossible.^[108]

p-adic numbers

Main article:p-adic number

The⁠ $p {\displaystyle p}$ ⁠-adic order $\nu _{p}(n)$ of an integer⁠ $n {\displaystyle n}$ ⁠ is the number of copies of⁠ $p {\displaystyle p}$ ⁠ in the prime factorization of⁠ $n {\displaystyle n}$ ⁠. The same concept can be extended from integers to rational numbers by defining the⁠ $p {\displaystyle p}$ ⁠-adic order of a fraction $m/n$ to be⁠ $\nu _{p}(m)-\nu _{p}(n)$ ⁠. The⁠ $p {\displaystyle p}$ ⁠-adic absolute value $|q|_{p}$ of any rational number⁠ $q {\displaystyle q}$ ⁠ is then defined as⁠ $\vert q\vert _{p}=p^{-\nu _{p}(q)}$ ⁠. Multiplying an integer by its⁠ $p {\displaystyle p}$ ⁠-adic absolute value cancels out the factors of⁠ $p {\displaystyle p}$ ⁠ in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their⁠ $p {\displaystyle p}$ ⁠-adic distance, the⁠ $p {\displaystyle p}$ ⁠-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of⁠ $p {\displaystyle p}$ ⁠. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form acomplete field, the rational numbers with the⁠ $p {\displaystyle p}$ ⁠-adic distance can be extended to a different complete field, the⁠ $p {\displaystyle p}$ ⁠-adic numbers.^[109]^[110]

This picture of an order, absolute value, and complete field derived from them can be generalized toalgebraic number fields and theirvaluations (certain mappings from themultiplicative group of the field to atotally ordered additive group, also called orders),absolute values (certain multiplicative mappings from the field to the real numbers, also callednorms),^[109] and places (extensions tocomplete fields in which the given field is adense set, also called completions).^[111] The extension from the rational numbers to thereal numbers, for instance, is a place in which the distance between numbers is the usualabsolute value of their difference. The corresponding mapping to an additive group would be thelogarithm of the absolute value, although this does not meet all the requirements of a valuation. According toOstrowski's theorem, up to a natural notion of equivalence, the real numbers and⁠ $p {\displaystyle p}$ ⁠-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers.^[109] Thelocal–global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.^[112]

Prime elements of a ring

Main articles:Prime element andIrreducible element

AllGaussian primes with norm squared less than 500

Acommutative ring is analgebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways,prime elements andirreducible elements. An element⁠ $p {\displaystyle p}$ ⁠ of a ring⁠ $R {\displaystyle R}$ ⁠ is called prime if it is nonzero, has nomultiplicative inverse (that is, it is not aunit), and satisfies the following requirement: whenever⁠ $p {\displaystyle p}$ ⁠ divides the product $x y {\displaystyle xy}$ of two elements of⁠ $R {\displaystyle R}$ ⁠, it also divides at least one of⁠ $x {\displaystyle x}$ ⁠ or⁠ $y {\displaystyle y}$ ⁠. An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set,

\{\dots ,-11,-7,-5,-3,-2,2,3,5,7,11,\dots \}\,.

In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold forunique factorization domains.^[113]

The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is theGaussian integers⁠ $\mathbb {Z} [i]$ ⁠, the ring ofcomplex numbers of the form $a+bi$ where⁠ $i {\displaystyle i}$ ⁠ denotes theimaginary unit and⁠ $a {\displaystyle a}$ ⁠ and⁠ $b {\displaystyle b}$ ⁠ are arbitrary integers. Its prime elements are known asGaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes $1+i$ and⁠ $1-i$ ⁠. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.^[114] This is a consequence ofFermat's theorem on sums of two squares,which states that an odd prime⁠ $p {\displaystyle p}$ ⁠ is expressible as the sum of two squares,⁠ $p=x^{2}+y^{2}$ ⁠, and therefore factorable as⁠ $p=(x+iy)(x-iy)$ ⁠, exactly when⁠ $p {\displaystyle p}$ ⁠ is 1 mod 4.^[115]

Prime ideals

Main article:Prime ideals

Not every ring is a unique factorization domain. For instance, in the ring of numbers $a+b{\sqrt {-5}}$ (for integers⁠ $a {\displaystyle a}$ ⁠ and⁠ $b {\displaystyle b}$ ⁠) the number $21 {\displaystyle 21}$ has two factorizations⁠ $21=3\cdot 7=(1+2{\sqrt {-5}})(1-2{\sqrt {-5}})$ ⁠, where none of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of anideal, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements.Prime ideals, which generalize prime elements in the sense that theprincipal ideal generated by a prime element is a prime ideal, are an important tool and object of study incommutative algebra,algebraic number theory andalgebraic geometry. The prime ideals of the ring of integers are the ideals⁠ $(0) {\displaystyle (0)}$ ⁠,⁠ $(2) {\displaystyle (2)}$ ⁠,⁠ $(3) {\displaystyle (3)}$ ⁠,⁠ $(5) {\displaystyle (5)}$ ⁠,⁠ $(7) {\displaystyle (7)}$ ⁠,⁠ $(11) {\displaystyle (11)}$ ⁠, ... The fundamental theorem of arithmetic generalizes to theLasker–Noether theorem, which expresses every ideal in aNoetherian commutative ring as an intersection ofprimary ideals, which are the appropriate generalizations ofprime powers.^[116]

Thespectrum of a ring is a geometric space whose points are the prime ideals of the ring.^[117]Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization orramification of prime ideals when lifted to anextension field, a basic problem of algebraic number theory, bears some resemblance withramification in geometry. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in thering of integers ofquadratic number fields can be used in provingquadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers.^[118] Early attempts to proveFermat's Last Theorem led toKummer's introduction ofregular primes, integer prime numbers connected with the failure of unique factorization in thecyclotomic integers.^[119] The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed byChebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.^[120]

Group theory

In the theory offinite groups theSylow theorems imply that, if a power of a prime number $p^{n}$ divides theorder of a group, then the group has a subgroup of order⁠ $p^{n}$ ⁠. ByLagrange's theorem, any group of prime order is acyclic group,and byBurnside's theorem any group whose order is divisible by only two primes issolvable.^[121]

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] other than the use of prime numbered gear teeth to distribute wear evenly.^[122] In particular, number theorists such asBritish mathematicianG. H. Hardy prided themselves on doing work that had absolutely no military significance.^[123]

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 ofpublic-key cryptography algorithms.^[33]These applications have led to significant study ofalgorithms for computing with prime numbers, and in particular ofprimality testing, methods for determining whether 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 calledfactorization algorithms. Prime numbers are also used in computing forchecksums,hash tables, andpseudorandom number generators.

Trial division

Main article:Trial division

The most basic method of checking the primality of a given integer⁠ $n {\displaystyle n}$ ⁠ is calledtrial division. This method divides⁠ $n {\displaystyle n}$ ⁠ by each integer from 2 up to thesquare 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 $n=a\cdot b$ ⁠, one of the two factors⁠ $a {\displaystyle a}$ ⁠ and⁠ $b {\displaystyle b}$ ⁠ is less than or equal to thesquare root of⁠ $n {\displaystyle n}$ ⁠. Another optimization is to check only primes as factors in this range.^[124] For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to⁠ ${\sqrt {37}}$ ⁠, 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 performsgrows exponentially as a function of the number of digits of these integers.^[125] 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.^[126]

Sieves

Animation of the sieve of Eratosthenes — Thesieve 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.^[127] The oldest known method for generating a list of primes is called the sieve of Eratosthenes.^[128] The animation shows an optimized variant of this method.^[129] Another more asymptotically efficient sieving method for the same problem is thesieve of Atkin.^[130] In advanced mathematics,sieve theory applies similar methods to other problems.^[131]

Primality testing versus primality proving

Some of the fastest modern tests for whether an arbitrary given number⁠ $n {\displaystyle n}$ ⁠ is prime areprobabilistic (orMonte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer.^[132] For instance theSolovay–Strassen primality test on a given number⁠ $p {\displaystyle p}$ ⁠ chooses a number⁠ $a {\displaystyle a}$ ⁠ randomly from 2 through $p-2$ and usesmodular exponentiation to check whether $a^{(p-1)/2}\pm 1$ 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.^[133] 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 $1/2^{n}$ ⁠. 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.^[134]A composite number that passes such a test is called apseudoprime.^[133]

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 bothdeterministic (non-random) algorithms, such as theAKS primality test,^[135]and randomizedLas Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations ofelliptic curve primality proving.^[132]When the elliptic curve method concludes that a number is prime, it providesprimality certificate that can be verified quickly.^[136]The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based onheuristic arguments rather than rigorous proofs. TheAKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice.^[137] 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, $\varepsilon$ is an arbitrarily small positive number, and log is thelogarithm to an unspecified base. Thebig O notation means that each time bound should be multiplied by aconstant 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	$O((\log n)^{6+\varepsilon })$		^[135]^[138]
Elliptic curve primality proving	1986	Las Vegas	$O((\log n)^{4+\varepsilon })$ heuristically		^[137]
Baillie–PSW primality test	1980	Monte Carlo	$O((\log n)^{2+\varepsilon })$		^[139]^[140]
Miller–Rabin primality test	1980	Monte Carlo	$O(k(\log n)^{2+\varepsilon })$	error probability $4^{-k}$	^[141]
Solovay–Strassen primality test	1977	Monte Carlo	$O(k(\log n)^{2+\varepsilon })$	error probability $2^{-k}$	^[141]

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, theLucas–Lehmer primality test can determine whether aMersenne number (one less than apower of two) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test.^[142] This is why since 1992 (as of October 2024^[update]) thelargestknown prime has always been a Mersenne prime.^[143] It is conjectured that there are infinitely many Mersenne primes.^[144]

The following table gives the largest known primes of various types. Some of these primes have been found usingdistributed computing. In 2009, theGreat Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits.^[145] TheElectronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.^[146]

Type	Prime	Number of decimal digits	Date	Found by
Mersenne prime	2^136,279,841 − 1	41,024,320	October 12, 2024^[1]	Luke Durant,Great Internet Mersenne Prime Search
Proth prime	10,223 × 2^31,172,165 + 1	9,383,761	October 31, 2016^[147]	Péter Szabolcs,PrimeGrid^[148]
factorial prime	208,003! − 1	1,015,843	July 2016	Sou Fukui^[149]
primorial prime^[e]	1,098,133# − 1	476,311	March 2012	James P. Burt,PrimeGrid^[151]
twin primes	2,996,863,034,895 × 2^1,290,000 ± 1	388,342	September 2016	Tom Greer,PrimeGrid^[152]

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 asfactorization of⁠ $n {\displaystyle n}$ ⁠. It is significantly more difficult than primality testing,^[153] and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division andPollard's rho algorithm can be used to find very small factors of⁠ $n {\displaystyle n}$ ⁠,^[126] andelliptic curve factorization can be effective when⁠ $n {\displaystyle n}$ ⁠ has factors of moderate size.^[154] Methods suitable for arbitrary large numbers that do not depend on the size of its factors include thequadratic sieve andgeneral number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including thespecial number field sieve.^[155] As of December 2019^[update] thelargest number known to have been factored by a general-purpose algorithm isRSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes.^[156]

Shor's algorithm can factor any integer in a polynomial number of steps on aquantum computer.^[157] However, current technology can only run this algorithm for very small numbers. As of October 2012^[update], the largest number that has been factored by a quantum computer running Shor's algorithm is 21.^[158]

Other computational applications

Severalpublic-key cryptography algorithms, such asRSA and theDiffie–Hellman key exchange, are based on large prime numbers (2048-bit primes are common).^[159] 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 $y {\displaystyle y}$ ⁠ (assumedcoprime) if only the product $x y {\displaystyle xy}$ is known.^[33] The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms formodular exponentiation (computing⁠ $a^{b}{\bmod {c}}$ ⁠), while the reverse operation (thediscrete logarithm) is thought to be a hard problem.^[160]

Prime numbers are frequently used forhash tables. For instance the original method of Carter and Wegman foruniversal hashing was based on computinghash functions by choosing randomlinear 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.^[161] As well as in the hash function, prime numbers are used for the hash table size inquadratic probing based hash tables to ensure that the probe sequence covers the whole table.^[162]

Somechecksum methods are based on the mathematics of prime numbers. For instance the checksums used inInternational 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.^[163] Another checksum method,Adler-32, uses arithmetic modulo 65521, the largest prime number less than⁠ $2^{16}$ ⁠.^[164] Prime numbers are also used inpseudorandom number generators includinglinear congruential generators^[165] and theMersenne Twister.^[166]

Other applications

Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, includingabstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so thatno three are in a line, or so that every triangle formed by three of the pointshas large area.^[167] Another example isEisenstein's criterion, a test for whether apolynomial is irreducible based on divisibility of its coefficients by a prime number and its square.^[168]

The concept of a 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, theprime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or afinite field with a prime number of elements, whence the name.^[169] 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, inknot theory, aprime knot is aknot that is indecomposable in the sense that it cannot be written as theconnected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.^[170] Theprime decomposition of 3-manifolds is another example of this type.^[171]

Beyond mathematics and computing, prime numbers have potential connections toquantum mechanics, and have been used metaphorically in the arts and literature. They have also been used inevolutionary biology to explain the life cycles ofcicadas.

Constructible polygons and polygon partitions

Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is aFermat prime.

Fermat primes are primes of the form

F_{k}=2^{2^{k}}+1,

with⁠ $k {\displaystyle k}$ ⁠ anonnegative integer.^[172] They are named afterPierre 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,^[173] but $F_{5}$ is composite and so are all other Fermat numbers that have been verified as of 2017.^[174] Aregular⁠ $n {\displaystyle n}$ ⁠-gon isconstructible using straightedge and compass if and only if the odd prime factors of⁠ $n {\displaystyle n}$ ⁠ (if any) are distinct Fermat primes.^[173] Likewise, a regular⁠ $n {\displaystyle n}$ ⁠-gon may be constructed using straightedge, compass, and anangle 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 distinctPierpont primes, primes of the form⁠ $2^{a}3^{b}+1$ ⁠.^[175]

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 apower of a prime number, but this is not known for other values of⁠ $n {\displaystyle n}$ ⁠.^[176]

Quantum mechanics

Beginning with the work ofHugh Montgomery andFreeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels ofquantum systems.^[177]^[178] Prime numbers are also significant inquantum information science, thanks to mathematical structures such asmutually unbiased bases andsymmetric informationally complete positive-operator-valued measures.^[179]^[180]

Biology

The evolutionary strategy used bycicadas of the genusMagicicada makes use of prime numbers.^[181] These insects spend most of their lives asgrubs 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 from synchronizing with these cycles.^[182]^[183] In contrast, the multi-year periods between flowering inbamboo plants are hypothesized to besmooth numbers, having only small prime numbers in their factorizations.^[184]

Arts and literature

Prime numbers have influenced many artists and writers. The Frenchcomposer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such asLa Nativité du Seigneur (1935) andQuatre études de rythme (1949–1950), 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".^[185]

In his science fiction novelContact, scientistCarl 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 astronomerFrank Drake in 1975.^[186] In the novelThe Curious Incident of the Dog in the Night-Time byMark 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 withAsperger syndrome.^[187] Prime numbers are used as a metaphor for loneliness and isolation in thePaolo Giordano novelThe Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.^[188]

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.^[29]
^^a ^bFor instance, Beiler writes that number theoristErnst Kummer loved hisideal numbers, closely related to the primes, "because they had not soiled themselves with any practical applications",^[31] and Katz writes thatEdmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such asgeometry that had already shown themselves to be useful.^[32]
^In this test, the $\pm 1$ 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 $\pm 1$ term is the (negated)Jacobi symbol, which can be calculated usingquadratic 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.^[136]
^Theprimorial function of⁠ $n {\displaystyle n}$ ⁠, denoted by⁠ $n\#$ ⁠, yields the product of the prime numbers up to⁠ $n {\displaystyle n}$ ⁠, and aprimorial prime is a prime of one of the forms⁠ $n\#\pm 1$ ⁠.^[150]

References

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External links

"Prime number".Encyclopedia of Mathematics.EMS Press. 2001 [1994].
Caldwell, Chris, ThePrime Pages atprimes.utm.edu.
Prime Numbers onIn Our Time at theBBC.
"Teacher package: Prime numbers" fromPlus, December 1, 2008, produced by the Millennium Mathematics Project at the University of Cambridge.

Generators and calculators

Prime factors calculator can factorize any positive integer up to 20 digits.
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Prime Numbers up to 1 trillion.Archived 2021-02-27 at theWayback Machine.

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v t e Number theory
Fields	Algebraic number theory (class field theory,non-abelian class field theory,Iwasawa theory,Iwasawa–Tate theory,Kummer theory) Analytic number theory (analytic theory of L-functions,probabilistic number theory,sieve theory) Geometric number theory Computational number theory Transcendental number theory Diophantine geometry (Arakelov theory,Hodge–Arakelov theory) Arithmetic combinatorics (additive number theory) Arithmetic geometry (anabelian geometry,p-adic Hodge theory) Arithmetic topology Arithmetic dynamics
Key concepts	Numbers 0 Natural numbers Unity Prime numbers Composite numbers Rational numbers Irrational numbers Algebraic numbers Transcendental numbers p-adic numbers (p-adic analysis) Arithmetic Modular arithmetic Chinese remainder theorem Arithmetic functions
Advanced concepts	Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Irrationality measure Simple continued fractions
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v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
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 Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

Prime number classes

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Patterns

k-tuples	Twin (p,p + 2) Triplet (p,p + 2 orp + 4,p + 6) Quadruplet (p,p + 2,p + 6,p + 8) Cousin (p,p + 4) Sexy (p,p + 6) Arithmetic progression (p + a·n,n = 0, 1, 2, 3, ...) Balanced (consecutivep − n,p,p + n)

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List of the first 1000 prime numbers

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