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A prime number (or a prime) is a
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
greater than 1 that is not a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of two smaller natural numbers. A natural number greater than 1 that is not prime is called a
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
. For example, 5 is prime because the only ways of writing it as a product, or , 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 in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
because of the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called
trial division Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer ''n'', the integer to be factored, can be divided by each number in turn t ...
, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the
Miller–Rabin primality test The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen prima ...
, which is fast but has a small chance of error, and the
AKS primality test The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists ...
, which always produces the correct answer in
polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as
Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
s. the
largest known prime number The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. A prime number is a pos ...
is a Mersenne prime with 24,862,048 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
, proven at the end of the 19th century, which says that the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of a randomly chosen large number being prime is inversely proportional to its number of digits, that is, to its
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
. Several historical questions regarding prime numbers are still unsolved. These include
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to h ...
, that every even integer greater than 2 can be expressed as the sum of two primes, and the
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in
information technology Information technology (IT) is the use of computers to create, process, store, retrieve, and exchange all kinds of data . and information. IT forms part of information and communications technology (ICT). An information technology syste ...
, such as
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
, which relies on the difficulty of factoring large numbers into their prime factors. In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, objects that behave in a generalized way like prime numbers include
prime element In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish ...
s and
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s.


Definition and examples

A
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
(1, 2, 3, 4, 5, 6, etc.) is called a ''prime number'' (or a ''prime'') if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called
composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s. In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange n dots into a rectangular grid that is more than one dot wide and more than one dot high. For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder). 1 is not prime, as it is specifically excluded in the definition. and are both composite. The
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s of a natural number n are the natural numbers that divide n evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive
divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
s, 1 and the number itself. Yet another way to express the same thing is that a number n is prime if it is greater than one and if none of the numbers 2, 3, \dots, n-1 divides n evenly. The first 25 prime numbers (all the prime numbers less than 100) 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 . No
even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 4 ...
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 an
odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...
, and is called an ''odd prime''. Similarly, when written in the usual
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
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. The set of all primes is sometimes denoted by \mathbf (a
boldface In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them. It is the equivalent of prosody stress in speech. Methods and use The most common methods in ...
capital ''P'') or by \mathbb (a
blackboard bold Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pro ...
capital P).


History

The
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchase ...
, from around 1550 BC, has
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from ea ...
expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from
ancient Greek mathematics Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
.
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
's ''
Elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of o ...
'' (c. 300 BC) proves the infinitude of primes and the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
, and shows how to construct a
perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. Th ...
from a
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
. Another Greek invention, the
Sieve of Eratosthenes In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime ...
, is still used to construct lists of Around 1000 AD, the
Islamic Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or '' Allah'') as it was revealed to Muhammad, the ma ...
mathematician
Ibn al-Haytham Ḥasan Ibn al-Haytham, Latinized as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
(Alhazen) found
Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations ...
, characterizing the prime numbers as the numbers 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. Another Islamic mathematician,
Ibn al-Banna' al-Marrakushi Ibn al‐Bannāʾ al‐Marrākushī ( ar, ابن البناء المراكشي), full name: Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi al-Marrakushi () (29 December 1256 – 31 July 1321), was a Moroccan polymath who was active as a mat ...
, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit.
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
brought the innovations from Islamic mathematics back to Europe. His book ''
Liber Abaci ''Liber Abaci'' (also spelled as ''Liber Abbaci''; "The Book of Calculation") is a historic 1202 Latin manuscript on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. ''Liber Abaci'' was among the first Western books to describ ...
'' (1202) was the first to describe
trial division Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer ''n'', the integer to be factored, can be divided by each number in turn t ...
for testing primality, again using divisors only up to the square root. In 1640
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
stated (without proof)
Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...
(later proved by
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
and Fermat also investigated the primality of the and Marin Mersenne studied the
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
s, prime numbers of the form 2^p-1 with p itself a prime.
Christian Goldbach Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After travelin ...
formulated
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to h ...
, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the
Euclid–Euler theorem The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematician ...
) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
to this area in his proofs of the infinitude of the primes and the
divergence of the sum of the reciprocals of the primes The sum of the reciprocals of all prime numbers diverges; that is: \sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-centur ...
\tfrac+\tfrac+\tfrac+\tfrac+\tfrac+\cdots. At the start of the 19th century, Legendre and Gauss conjectured that as x tends to infinity, the number of primes up to x is
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
to x/\log x, where \log x is the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of x. A weaker consequence of this high density of primes was
Bertrand's postulate In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with :n < p < 2n - 2. A less restrictive formulation is: for every n > 1, there is always ...
, that for every n > 1 there is a prime between n and 2n, proved in 1852 by
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...
. Ideas of
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
. Another important 19th century result was
Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...
, that certain
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s contain infinitely many primes. Many mathematicians have worked on
primality test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating wheth ...
s for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include
Pépin's test In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin. Description of the test Let F_n ...
for Fermat numbers (1877), Proth's theorem (c. 1878), the
Lucas–Lehmer primality test In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1876 and subsequently improved by Derrick Henry Lehmer in the 1930s. The test The Lucas–Lehmer test ...
(originated 1856), and the generalized
Lucas primality test In computational number theory, the Lucas test is a primality test for a natural number ''n''; it requires that the prime factors of ''n'' − 1 be already known. It is the basis of the Pratt certificate that gives a concise verification th ...
. Since 1951 all the largest known primes have been found using these tests on
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These prog ...
s. The search for ever larger primes has generated interest outside mathematical circles, through the
Great Internet Mersenne Prime Search The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client a ...
and other
distributed computing A distributed system is a system whose components are located on different networked computers, which communicate and coordinate their actions by passing messages to one another from any system. Distributed computing is a field of computer sc ...
projects. The idea that prime numbers had few applications outside of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
was shattered in the 1970s when
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
and the RSA cryptosystem were invented, using prime numbers as their basis. The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the
Green–Tao theorem In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number ''k'', there exist arith ...
(2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many
prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ W ...
s of bounded size., pp. 18, 47.


Primality of one

Most early Greeks did not even consider 1 to be a number, For a selection of quotes from and about the ancient Greek positions on the status of 1 and 2, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6. so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including
Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works ''Introduction to Arithmetic'' and '' Manual of Harmonics'' in Greek. He was born i ...
,
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
,
Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, '' magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the ...
, and
Cassiodorus Magnus Aurelius Cassiodorus Senator (c. 485 – c. 585), commonly known as Cassiodorus (), was a Roman statesman, renowned scholar of antiquity, and writer serving in the administration of Theodoric the Great, king of the Ostrogoths. ''Senator' ...
also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number. By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number. In the mid-18th century
Christian Goldbach Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After travelin ...
listed 1 as prime in his correspondence with
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
; however, Euler himself did not consider 1 to be prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956. If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with any number of copies of 1. Similarly, the
sieve of Eratosthenes In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime ...
would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In o ...
or for the sum of divisors function are different for prime numbers than they are for 1. By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
".


Elementary properties


Unique factorization

Writing a number as a product of prime numbers is called a ''prime factorization'' of the number. For example: :\begin 34866 &= 2\times 3\times 3\times 13 \times 149\\ &=2\times 3^2\times 13 \times 149. \end The terms in the product are called ''prime factors''. The same prime factor may occur more than once; this example has two copies of the prime factor 3. When a prime occurs multiple times,
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, 3^2 denotes the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
or second power of 3. The central importance of prime numbers to number theory and mathematics in general stems from the ''fundamental theorem of arithmetic''. This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly, 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. So, although there are many different ways of finding a factorization using an
integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are ...
algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers. Some proofs of the uniqueness of prime factorizations are based on
Euclid's lemma In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as we ...
: If p is a prime number and p divides a product ab of integers a and b, then p divides a or p divides b (or both). Conversely, if a number p has the property that when it divides a product it always divides at least one factor of the product, then p must be prime.


Infinitude

There are
infinitely Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
many prime numbers. Another way of saying this is that the sequence :2, 3, 5, 7, 11, 13, ... of prime numbers never ends. This statement is referred to as ''Euclid's theorem'' in honor of the ancient Greek mathematician
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by
Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
, Goldbach's
proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
based on
Fermat number In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
s, Furstenberg's proof using general topology, and Kummer's elegant proof. Euclid's proof shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add 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 has a prime factorization : N = p'_1\cdot p'_2\cdots p'_m with one or more prime factors. N is evenly divisible by each of these factors, but 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 can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes. The numbers formed by adding one to the products of the smallest primes are called Euclid numbers. 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

There is no known efficient formula for primes. For example, there is no non-constant
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
, even in several variables, that takes ''only'' prime values. However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on
Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations ...
and generates the number 2 many times and all other primes exactly once. There is also a set of
Diophantine equations In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to ...
in nine variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its ''positive'' values are prime. Other examples of prime-generating formulas come from Mills' theorem and a theorem of
Wright Wright is an occupational surname originating in England. The term 'Wright' comes from the circa 700 AD Old English word 'wryhta' or 'wyrhta', meaning worker or shaper of wood. Later it became any occupational worker (for example, a shipwright is ...
. These assert that there are real constants A>1 and \mu such that :\left \lfloor A^\right \rfloor \text \left \lfloor 2^ \right \rfloor are prime for any natural number n in the first formula, and any number of exponents in the second formula. Here \lfloor \cdot \rfloor represents the
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least inte ...
, 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 or \mu.


Open questions

Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of
Landau's problems At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau ...
from 1912 are still unsolved. One of them is
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to h ...
, which asserts that every even integer n greater than 2 can be written as a sum of two primes. , this conjecture has been verified for all numbers up to n=4\cdot 10^. Weaker statements than this have been proven, for example,
Vinogradov's theorem In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a rep ...
says that every sufficiently large odd integer can be written as a sum of three primes.
Chen's theorem In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). History The theorem was first stated by Chinese mathemat ...
says that every sufficiently large even number can be expressed as the sum of a prime and a
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime n ...
(the product of two primes). Also, any even integer greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called
additive number theory Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigro ...
. Another type of problem concerns
prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ W ...
s, 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. However, large prime gaps occur much earlier than this argument shows. For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than 8!=40320. It is conjectured that there are infinitely many
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
s, pairs of primes with difference 2; this is the
twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
. Polignac's conjecture states more generally that for every positive integer k, there are infinitely many pairs of consecutive primes that differ by 2k., Gaps between primes, pp. 186–192. Andrica's conjecture, Brocard's conjecture,, p. 183. Legendre's conjecture, Note that Chan lists Legendre's conjecture as "Sierpinski's Postulate". and
Oppermann's conjecture Oppermann's conjecture is an unsolved problem in mathematics on the distribution of prime numbers.. It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's conjecture. It is named after Danish mathem ...
all suggest that the largest gaps between primes from 1 to n should be at most approximately \sqrt, a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at O((\log n)^2). Prime gaps can be generalized to prime k-tuples, patterns in the differences between more than two prime numbers. Their infinitude and density are the subject of the
first Hardy–Littlewood conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...
, which can be motivated by the
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.


Analytic properties

Analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...
studies number theory through the lens of
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s,
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
,
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mat ...
, and the related mathematics of the infinite and
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally ref ...
. This area of study began with
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
and his first major result, the solution to the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
. The problem asked for the value of the infinite sum 1+\tfrac+\tfrac+\tfrac+\dots, which today can be recognized as the value \zeta(2) of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis. Euler showed that \zeta(2)=\pi^2/6. The reciprocal of this number, 6/\pi^2, is the limiting probability that two random numbers selected uniformly from a large range are
relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
(have no factors in common). The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
, but no efficient formula for the n-th prime is known.
Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...
, in its basic form, asserts that linear polynomials :p(n) = a + bn with relatively prime integers a and 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 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 of reciprocals of primes, :\frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p. Euler showed that, for any arbitrary
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
x, there exists a prime p for which this sum is bigger than x. 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. The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum :\frac 1 + \frac 1 + \frac 1 + \cdots + \frac 1 does not grow to infinity as n goes to infinity (see the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
). In this sense, prime numbers occur more often than squares of natural numbers, although both sets are infinite.
Brun's theorem In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by ''B''2 . Brun's theorem was proved by ...
states that the sum of the reciprocals of
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
s, : \left( \right) + \left( \right) + \left( \right) + \cdots, is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the
twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
, that there exist infinitely many twin primes.


Number of primes below a given bound

The
prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory i ...
\pi(n) is defined as the number of primes not greater than n. For example, \pi(11)=5, since there are five primes less than or equal to 11. Methods such as the
Meissel–Lehmer algorithm The Meissel–Lehmer algorithm (after Ernst Meissel and Derrick Henry Lehmer) is an algorithm that computes exact values of the prime-counting function. Description The problem of counting the exact number of primes less than or equal to x, wit ...
can compute exact values of \pi(n) faster than it would be possible to list each prime up to n. The
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
states that \pi(n) is asymptotic to n/\log n, which is denoted as : \pi(n) \sim \frac, and means that the ratio of \pi(n) to the right-hand fraction approaches 1 as n grows to infinity.
p. 10
This implies that the likelihood that a randomly chosen number less than n is prime is (approximately) inversely proportional to the number of digits in n. It also implies that the nth prime number is proportional to n\log n and therefore that the average size of a prime gap is proportional to \log n.,
Large gaps between consecutive primes
, pp. 78–79.
A more accurate estimate for \pi(n) is given by the
offset logarithmic integral In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
:\pi(n)\sim \operatorname(n) = \int_2^n \frac.


Arithmetic progressions

An
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. This difference is called the modulus of the progression. For example, :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 and modulus q are relatively prime. If they are relatively prime,
Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is al ...
asserts that the progression contains infinitely many primes. The
Green–Tao theorem In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number ''k'', there exist arith ...
shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.


Prime values of quadratic polynomials

Euler noted that the function :n^2 - n + 41 yields prime numbers for 1\le n\le 40, although composite numbers appear among its later values. The search for an explanation for this phenomenon led to the deep
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ...
of
Heegner number In number theory, a Heegner number (as termed by Conway and Guy) is a square-free positive integer ''d'' such that the imaginary quadratic field \Q\left sqrt\right/math> has class number 1. Equivalently, its ring of integers has unique factori ...
s and the
class number problem In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having ...
. The
Hardy-Littlewood conjecture F The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's ''Mathematical Games'' column in ''Scientific American'' a short time late ...
predicts the density of primes among the values of
quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomi ...
s with integer
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves va ...
s in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values. The
Ulam spiral The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's ''Mathematical Games'' column in ''Scientific American'' a short time late ...
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.


Zeta function and the Riemann hypothesis

One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
\zeta(s) are located. This function is an
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
on the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. For complex numbers s with real part greater than one it equals both an
infinite sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
over all integers, and an
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound. ...
over the prime numbers, :\zeta(s)=\sum_^\infty \frac=\prod_ \frac 1 . This equality between a sum and a product, discovered by Euler, is called an
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard E ...
. 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. 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 the harmonic series 1+\tfrac+\tfrac+\dots) while the product would be finite, a contradiction. The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
equal to 1/2. The original proof of the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1,
p. 18.
/ref> although other more elementary proofs have been found. The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term. In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the
asymptotic distribution In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing ap ...
of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of x for intervals near a number x).


Abstract algebra


Modular arithmetic and finite fields

Modular arithmetic modifies usual arithmetic by only using the numbers \, for a natural number n called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by n. 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. Equality of integers corresponds to ''congruence'' in modular arithmetic: x and y are congruent (written x\equiv y mod n) when they have the same remainder after division by n. However, in this system of numbers,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
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, because
clearing denominators In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. Example Co ...
by multiplying both sides by 3 gives the valid formula 2\equiv 9\bmod. However, with the composite modulus 6, division by 3 is impossible. There is no valid solution to 2/3\equiv x\bmod: 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 of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, the ability to perform division means that modular arithmetic modulo a prime number forms a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
or, more specifically, a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, sub ...
, while other moduli only give a ring but not a field. Several theorems about primes can be formulated using modular arithmetic. For instance,
Fermat's little theorem Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as : a^p \equiv a \pmod p. For example, if = ...
states that if a\not\equiv 0 (mod p), then a^\equiv 1 (mod p). Summing this over all choices of a gives the equation :\sum_^ a^ \equiv (p-1) \cdot 1 \equiv -1 \pmod p, valid whenever p is prime. Giuga's conjecture says that this equation is also a sufficient condition for p to be prime.
Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations ...
says that an integer p>1 is prime if and only if the
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) ...
(p-1)! is congruent to -1 mod p. For a composite this cannot hold, since one of its factors divides both and (n-1)!, and so (n-1)!\equiv -1 \pmod is impossible.


''p''-adic numbers

The p-adic order \nu_p(n) of an integer n is the number of copies of p in the prime factorization of n. The same concept can be extended from integers to rational numbers by defining the p-adic order of a fraction m/n to be \nu_p(m)-\nu_p(n). The p-adic absolute value , q, _p of any rational number q is then defined as , q, _p=p^. Multiplying an integer by its p-adic absolute value cancels out the factors of 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-adic distance, the 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. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a
complete field In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the ''p''-adic numbers). Constructions ...
, the rational numbers with the p-adic distance can be extended to a different complete field, the p-adic numbers. This picture of an order, absolute value, and complete field derived from them can be generalized to
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s and their valuations (certain mappings from the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings from the field to the real numbers, also called norms), See also p. 64. and places (extensions to
complete field In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the ''p''-adic numbers). Constructions ...
s in which the given field is a
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ...
, also called completions). The extension from the rational numbers to the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s, for instance, is a place in which the distance between numbers is the usual
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of their difference. The corresponding mapping to an additive group would be the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
of the absolute value, although this does not meet all the requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and p-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers. The
local-global principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each ...
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.


Prime elements in rings

A
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
is an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, ''prime elements'' and ''irreducible elements''. An element p of a ring R is called prime if it is nonzero, has no
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
(that is, it is not a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
), and satisfies the following requirement: whenever p divides the product xy of two elements of R, it also divides at least one of x or 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, :\\, . In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
s. The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s \mathbb /math>, the ring of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of the form a+bi where i denotes the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and ...
and a and b are arbitrary integers. Its prime elements are known as
Gaussian prime In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathb ...
s. 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. This is a consequence of
Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true ar ...
, which states that an odd prime 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 is 1 mod 4.


Prime ideals

Not every ring is a unique factorization domain. For instance, in the ring of numbers a+b\sqrt (for integers a and b) the number 21 has two factorizations 21=3\cdot7=(1+2\sqrt)(1-2\sqrt), where neither 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 an ideal, 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 the
principal ideal In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it ...
generated by a prime element is a prime ideal, are an important tool and object of study in
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
,
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ...
and
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the
Lasker–Noether theorem In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related ...
, which expresses every ideal in a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
as an intersection of
primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. Fo ...
s, which are the appropriate generalizations of
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
s. The
spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
is a geometric space whose points are the prime ideals of the ring.
Arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...
also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an
extension field In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often denot ...
of
quadratic number field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
s can be used in proving
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
led to
Kummer Kummer is a German surname. Notable people with the surname include: * Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873—1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chri ...
's introduction of
regular prime In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli nu ...
s, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers. The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by
Chebotarev's density theorem Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension ''K'' of the field \mathbb of rational numbers. Generally speaking, a prime integer will factor into several idea ...
, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.


Group theory

In the theory of
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
s the
Sylow theorems In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...
imply that, if a power of a prime number p^n divides the
order of a group In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subg ...
, then the group has a subgroup of order p^n. By Lagrange's theorem, any group of prime order is a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
, and by
Burnside's theorem In mathematics, Burnside's theorem in group theory states that if ''G'' is a finite group of order p^a q^b where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers, then ''G'' is solvable. Hence each non-Abelian fin ...
any group whose order is divisible by only two primes is solvable.


Computational methods

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 other than the use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as
British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity and common culture * British English, ...
mathematician
G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
prided themselves on doing work that had absolutely no military significance. This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
algorithms. These applications have led to significant study of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s for computing with prime numbers, and in particular of
primality test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating wheth ...
ing, 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 called
factorization In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kin ...
algorithms. Prime numbers are also used in computing for
checksum A checksum is a small-sized block of data derived from another block of digital data for the purpose of detecting errors that may have been introduced during its transmission or storage. By themselves, checksums are often used to verify data ...
s,
hash table In computing, a hash table, also known as hash map, is a data structure that implements an associative array or dictionary. It is an abstract data type that maps keys to values. A hash table uses a hash function to compute an ''index'', ...
s, and
pseudorandom number generator A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
s.


Trial division

The most basic method of checking the primality of a given integer n is called ''
trial division Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer ''n'', the integer to be factored, can be divided by each number in turn t ...
''. This method divides n by each integer from 2 up to the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of n. Any such integer dividing n evenly establishes 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 and b is less than or equal to the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
of n. Another optimization is to check only primes as factors in this range. For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to \sqrt, which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers. 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.
p. 220


Sieves

Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed. The oldest method for generating a list of primes is called the sieve of Eratosthenes. The animation shows an optimized variant of this method. Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin. In advanced mathematics, sieve theory applies similar methods to other problems.


Primality testing versus primality proving

Some of the fastest modern tests for whether an arbitrary given number n is prime are
probabilistic Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
(or
Monte Carlo Monte Carlo (; ; french: Monte-Carlo , or colloquially ''Monte-Carl'' ; lij, Munte Carlu ; ) is officially an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is ...
) algorithms, meaning that they have a small random chance of producing an incorrect answer. For instance the
Solovay–Strassen primality test The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen in 1977, is a probabilistic test to determine if a number is composite or probably prime. The idea behind the test was discovered by M. M. Artjuhov in 1967 ...
on a given number p chooses a number a randomly from 2 through p-2 and uses modular exponentiation to check whether a^\pm 1 is divisible by p. If so, it answers yes and otherwise it answers no. If p really is prime, it will always answer yes, but if p is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2. If this test is repeated 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. A composite number that passes such a test is called a
pseudoprime A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime t ...
. In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
(non-random) algorithms, such as the
AKS primality test The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists ...
, and randomized
Las Vegas algorithm In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on th ...
s where the random choices made by the algorithm do not affect its final answer, such as some variations of
elliptic curve primality proving In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 a ...
. When the elliptic curve method concludes that a number is prime, it provides
primality certificate In mathematics and computer science, a primality certificate or primality proof is a succinct, formal proof that a number is prime. Primality certificates allow the primality of a number to be rapidly checked without having to run an expensive or un ...
that can be verified quickly. The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. The
AKS primality test The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists ...
has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice. 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. The following table lists some of these tests. Their running time is given in terms of n, the number to be tested and, for probabilistic algorithms, the number k of tests performed. Moreover, \varepsilon is an arbitrarily small positive number, and log is the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
to an unspecified base. The
big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters n and k.


Special-purpose algorithms and the largest known prime

In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the
Lucas–Lehmer primality test In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1876 and subsequently improved by Derrick Henry Lehmer in the 1930s. The test The Lucas–Lehmer test ...
can determine whether a
Mersenne number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
(one less than a
power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negativ ...
) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test. This is why since 1992 () the largest ''known'' prime has always been a Mersenne prime. It is conjectured that there are infinitely many Mersenne primes. The following table gives the largest known primes of various types. Some of these primes have been found using
distributed computing A distributed system is a system whose components are located on different networked computers, which communicate and coordinate their actions by passing messages to one another from any system. Distributed computing is a field of computer sc ...
. In 2009, the
Great Internet Mersenne Prime Search The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client a ...
project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits. The
Electronic Frontier Foundation The Electronic Frontier Foundation (EFF) is an international non-profit digital rights group based in San Francisco, California. The foundation was formed on 10 July 1990 by John Gilmore, John Perry Barlow and Mitch Kapor to promote Internet c ...
also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.


Integer factorization

Given a composite integer n, the task of providing one (or all) prime factors is referred to as ''factorization'' of n. It is significantly more difficult than primality testing, and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and
Pollard's rho algorithm Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of th ...
can be used to find very small factors of n, and
elliptic curve factorization The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the thi ...
can be effective when n has factors of moderate size. Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the
quadratic sieve The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considera ...
and
general number field sieve In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than . Heuristically, its complexity for factoring an integer (consisting of bits) is of the form :\exp\left( ...
. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the
special number field sieve In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integ ...
. the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes.
Shor's algorithm Shor's algorithm is a quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor. On a quantum computer, to factor an integer N , Shor's algorithm runs in polynomial ...
can factor any integer in a polynomial number of steps on a
quantum computer Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
. However, current technology can only run this algorithm for very small numbers. the largest number that has been factored by a quantum computer running Shor's algorithm is 21.


Other computational applications

Several
public-key cryptography Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
algorithms, such as RSA and the
Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ...
, are based on large prime numbers (2048- bit primes are common). RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing a^b\bmod), while the reverse operation (the
discrete logarithm In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b'k'' can be defined for all integers ''k'', and the discrete logarithm log'' ...
) is thought to be a hard problem. Prime numbers are frequently used for
hash table In computing, a hash table, also known as hash map, is a data structure that implements an associative array or dictionary. It is an abstract data type that maps keys to values. A hash table uses a hash function to compute an ''index'', ...
s. For instance the original method of Carter and Wegman for
universal hashing In mathematics and computing, universal hashing (in a randomized algorithm or data structure) refers to selecting a hash function at random from a family of hash functions with a certain mathematical property (see definition below). This guarante ...
was based on computing
hash function A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called ''hash values'', ''hash codes'', ''digests'', or simply ''hashes''. The values are usually ...
s by choosing random
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. For dist ...
s modulo large prime numbers. Carter and Wegman generalized this method to k-independent hashing by using higher-degree polynomials, again modulo large primes. As well as in the hash function, prime numbers are used for the hash table size in
quadratic probing Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking the original hash index and adding successive values of an arbitrary quadratic polynomial un ...
based hash tables to ensure that the probe sequence covers the whole table. Some
checksum A checksum is a small-sized block of data derived from another block of digital data for the purpose of detecting errors that may have been introduced during its transmission or storage. By themselves, checksums are often used to verify data ...
methods are based on the mathematics of prime numbers. For instance the checksums used in
International Standard Book Number The International Standard Book Number (ISBN) is a numeric commercial book identifier that is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency. An ISBN is assigned to each separate edition an ...
s 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. Another checksum method,
Adler-32 Adler-32 is a checksum algorithm written by Mark Adler in 1995, modifying Fletcher's checksum. Compared to a cyclic redundancy check of the same length, it trades reliability for speed (preferring the latter). Adler-32 is more reliable than Fletche ...
, uses arithmetic modulo 65521, the largest prime number less than 2^. Prime numbers are also used in
pseudorandom number generator A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
s including
linear congruential generator A linear congruential generator (LCG) is an algorithm that yields a sequence of pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom number generat ...
s and the
Mersenne Twister The Mersenne Twister is a general-purpose pseudorandom number generator (PRNG) developed in 1997 by and . Its name derives from the fact that its period length is chosen to be a Mersenne prime. The Mersenne Twister was designed specifically to r ...
.


Other applications

Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area. Another example is
Eisenstein's criterion In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with ...
, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square. 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, the
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, sub ...
with a prime number of elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in
knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
, a
prime knot In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non- trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be c ...
is a
knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
that is indecomposable in the sense that it cannot be written as the
connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifica ...
of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The prime decomposition of 3-manifolds is another example of this type. Beyond mathematics and computing, prime numbers have potential connections to
quantum mechanic Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
s, and have been used metaphorically in the arts and literature. They have also been used in
evolutionary biology Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life fo ...
to explain the life cycles of
cicada The cicadas () are a superfamily, the Cicadoidea, of insects in the order Hemiptera (true bugs). They are in the suborder Auchenorrhyncha, along with smaller jumping bugs such as leafhoppers and froghoppers. The superfamily is divided into ...
s.


Constructible polygons and polygon partitions

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...
s are primes of the form :F_k = 2^+1, with k a
nonnegative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
. They are named after
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, but F_5 is composite and so are all other Fermat numbers that have been verified as of 2017. A regular n-gon is constructible using straightedge and compass if and only if the odd prime factors of n (if any) are distinct Fermat primes. Likewise, a regular n-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of n are any number of copies of 2 or 3 together with a (possibly empty) set of distinct
Pierpont prime In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who us ...
s, primes of the form 2^a3^b+1. It is possible to partition any convex polygon into n smaller convex polygons of equal area and equal perimeter, when n is a power of a prime number, but this is not known for other values of n.


Quantum mechanics

Beginning with the work of Hugh Montgomery and
Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum m ...
in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of
quantum system Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
s. Prime numbers are also significant in
quantum information science Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p ...
, thanks to mathematical structures such as
mutually unbiased bases In quantum information theory, mutually unbiased bases in Hilbert space C''d'' are two orthonormal bases \ and \ such that the square of the magnitude of the inner product between any basis states , e_j\rangle and , f_k\rangle equals the invers ...
and symmetric informationally complete positive-operator-valued measures.


Biology

The evolutionary strategy used by
cicada The cicadas () are a superfamily, the Cicadoidea, of insects in the order Hemiptera (true bugs). They are in the suborder Auchenorrhyncha, along with smaller jumping bugs such as leafhoppers and froghoppers. The superfamily is divided into ...
s of the genus '' Magicicada'' makes use of prime numbers. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles. In contrast, the multi-year periods between flowering in
bamboo Bamboos are a diverse group of evergreen perennial flowering plants making up the subfamily Bambusoideae of the grass family Poaceae. Giant bamboos are the largest members of the grass family. The origin of the word "bamboo" is uncertain, b ...
plants are hypothesized to be
smooth number In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 5 ...
s, having only small prime numbers in their factorizations.


Arts and literature

Prime numbers have influenced many artists and writers. The French
composer A composer is a person who writes music. The term is especially used to indicate composers of Western classical music, or those who are composers by occupation. Many composers are, or were, also skilled performers of music. Etymology and Def ...
Olivier Messiaen Olivier Eugène Prosper Charles Messiaen (, ; ; 10 December 1908 – 27 April 1992) was a French composer, organist, and ornithologist who was one of the major composers of the 20th century. His music is rhythmically complex; harmonically ...
used prime numbers to create ametrical music through "natural phenomena". In works such as '' La Nativité du Seigneur'' (1935) and '' Quatre études de rythme'' (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations". In his science fiction novel ''
Contact Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * ...
'', scientist
Carl Sagan Carl Edward Sagan (; ; November 9, 1934December 20, 1996) was an American astronomer, planetary scientist, cosmologist, astrophysicist, astrobiologist, author, and science communicator. His best known scientific contribution is research on e ...
suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer
Frank Drake Frank Donald Drake (May 28, 1930 – September 2, 2022) was an American astrophysicist and astrobiologist. He began his career as a radio astronomer, studying the planets of the Solar System and later pulsars. Drake expanded his interests t ...
in 1975. In the novel '' The Curious Incident of the Dog in the Night-Time'' by
Mark Haddon Mark Haddon (born 28 October 1962) is an English novelist, best known for '' The Curious Incident of the Dog in the Night-Time'' (2003). He won the Whitbread Award, the Dolly Gray Children's Literature Award, Guardian Prize, and a Commonwealth ...
, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with
Asperger syndrome Asperger syndrome (AS), also known as Asperger's, is a former neurodevelopmental disorder characterized by significant difficulties in social interaction and nonverbal communication, along with restricted and repetitive patterns of behavi ...
. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel ''
The Solitude of Prime Numbers ''The Solitude of Prime Numbers'' ( it, La solitudine dei numeri primi) is a 2010 Italian drama film based on the novel of the same name by Paolo Giordano. Directed by Saverio Costanzo, the film was nominated for the Golden Lion at the 67th V ...
'', in which they are portrayed as "outsiders" among integers.


Notes


References


External links

* * Caldwell, Chris, The
Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...
a
primes.utm.edu
*

from Plus, the free online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.


Generators and calculators



can factorize any positive integer up to 20 digits.
Fast Online primality test with factorization
makes use of the Elliptic Curve Method (up to thousand-digits numbers, requires Java).
Huge database of prime numbers


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