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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 In mathematics, a divisor of an integer n, also called a factor ...
. 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, "Ma ...
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'' ...
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 prim ...
, 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 Determinism is a philosophical view, where all events are determined completely by previously existing causes. Determi ...
, 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 ...
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 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 ...
. 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 hold ...
, 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 p ...
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 Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic or analytical can also have the following meanings: Chemistry * ...
or
algebraic Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
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 a ...
, 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 ter ...
, 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 pri ...
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 wi ...
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 In mathematics, a divisor of an integer n, also called a factor ...
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 \\ 41 ...
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 numeral ...
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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 W ...
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 Scotland, Scottish antiquarian, who ...
, 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 mathem ...
.
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 ...
's '' Elements'' (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. ...
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 17 ...
. 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 n ...
, 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 pr ...
(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 of m ...
, 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 math ...
, 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 Wester ...
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 describe ...
'' (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 ...
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 ma ...
and Fermat also investigated the primality of the and
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
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 17 ...
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 traveling ...
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 hold ...
, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from
mathematical analysis 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 context ...
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 alw ...
, 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 ...
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 In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
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 ...
, 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 whet ...
s for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877),
Proth's theorem In number theory, Proth's theorem is a primality test for Proth numbers. It states that if ''p'' is a Proth number, of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2''n'', and if there exists an
(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 te ...
(originated 1856), and the generalized Lucas primality test. 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 sci ...
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 a ...
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, Iamblichus, Boethius, and Cassiodorus also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The Mathematics in medieval Islam, 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 traveling ...
listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956. If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with any number of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the Sum-of-divisors function, 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 (ring theory), unit".


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 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 (algebra), square 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 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: 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 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 ...
, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an mathematical analysis, analytical proof by Euler, Christian Goldbach, Goldbach's Fermat number#Basic properties, proof based on Fermat numbers, Hillel Furstenberg, Furstenberg's Furstenberg's proof of the infinitude of primes, proof using general topology, and Ernst Kummer, Kummer's elegant proof. Euclid's theorem, Euclid's proof shows that every finite set, 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, 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 of m ...
and generates the number 2 many times and all other primes exactly once. There is also a set of Diophantine equations in nine variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its ''positive'' values are prime. Other examples of prime-generating formulas come from Mills' theorem and a theorem of E. M. WWright. 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, 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 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 hold ...
, 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 says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). Also, any even integer greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory. 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 p ...
s, pairs of primes with difference 2; this is the twin prime conjecture. 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 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-tuple, 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, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.


Analytic properties

Analytic number theory studies number theory through the lens of continuous functions, Limit (mathematics), limits, Series (mathematics), infinite series, and the related mathematics of the infinite and infinitesimal. This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem. 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. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
. 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 coprime integers, relatively prime (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 primes, 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 ...
, 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

Proof that the sum of the reciprocals of the primes diverges, Euler's proof that there are infinitely many primes considers the sums of Multiplicative inverse, 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 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' theorems, 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). In this sense, prime numbers occur more often than squares of natural numbers, although both sets are infinite. Brun's theorem 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 p ...
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, that there exist infinitely many twin primes.


Number of primes below a given bound

The prime-counting function \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 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 convergent sequence, 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 :\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 Modular arithmetic, 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 ...
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 of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients 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 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 In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
, which asks where the zero of a function, zeros of the Riemann zeta function \zeta(s) are located. This function is an analytic function on the complex numbers. For complex numbers s with real part greater than one it equals both an series (mathematics), infinite sum over all integers, and an infinite product 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. 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 (mathematics), harmonic series 1+\tfrac+\tfrac+\dots) while the product would be finite, a contradiction. The Riemann hypothesis states that the zero of a function, zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the
prime number theorem 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 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 (mathematics), division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number 7 as modulus, division by 3 is possible: 2/3\equiv 3\bmod, because clearing denominators 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 ter ...
, the ability to perform division means that modular arithmetic modulo a prime number forms a field (mathematics), field or, more specifically, a finite field, while other moduli only give a ring (mathematics), 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 of m ...
says that an integer p>1 is prime if and only if the factorial (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, 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, the rational numbers with the p-adic distance can be extended to a different complete field, the p-adic number, p-adic numbers. This picture of an order, absolute value, and complete field derived from them can be generalized to algebraic number fields and their Valuation (algebra), valuations (certain mappings from the multiplicative group of the field to a totally ordered group, totally ordered additive group, also called orders), Absolute value (algebra), absolute values (certain multiplicative mappings from the field to the real numbers, also called norms), See also p. 64. and places (extensions to complete fields in which the given field is a dense set, also called completions). The extension from the rational numbers to the real numbers, for instance, is a place in which the distance between numbers is the usual absolute value of their difference. The corresponding mapping to an additive group would be the
logarithm 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 ...
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 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 is an algebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, ''prime elements'' and ''irreducible elements''. An element p of a ring R is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a Unit (ring theory), unit), 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 domains. The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers \mathbb[i], the ring of complex numbers of the form a+bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes 1+i and 1-i. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not. This is a consequence of Fermat's theorem on sums of two squares, 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 (ring theory), 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 generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, number theory, algebraic number theory and algebraic geometry. 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, which expresses every ideal in a Noetherian ring, Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers. The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or Splitting of prime ideals in Galois extensions, ramification of prime ideals when lifted to an field extension, extension field, a basic problem of algebraic number theory, bears some resemblance with ramified cover, 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 of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Ernst Kummer, Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic field, 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, 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 groups the Sylow theorems imply that, if a power of a prime number p^n divides the order of a group, then the group has a subgroup of order p^n. By Lagrange's theorem (group theory), Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two primes is solvable group, 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 United Kingdom, British mathematician G. H. Hardy 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 a ...
algorithms. These applications have led to significant study of algorithms 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 whet ...
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 integer factorization, factorization algorithms. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators.


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 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 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 exponential growth, 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 algorithm, probabilistic (or Monte Carlo algorithm, Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer. For instance the Solovay–Strassen primality test 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. 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 algorithm, deterministic (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 Determinism is a philosophical view, where all events are determined completely by previously existing causes. Determi ...
, and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving. When the elliptic curve method concludes that a number is prime, it provides primality certificate 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 Determinism is a philosophical view, where all events are determined completely by previously existing causes. Determi ...
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 ...
to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters n 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 te ...
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) 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, 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 sci ...
. 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 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 can be used to find very small factors of n, and elliptic curve factorization 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 and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve. the Integer factorization records, 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 can factor any integer in a polynomial number of steps on a quantum computer. 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 a ...
algorithms, such as RSA (algorithm), RSA and the Diffie–Hellman key exchange, 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) 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) is thought to be a hard problem. Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to k-independent hashing, 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 based hash tables to ensure that the probe sequence covers the whole table. Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than 2^. Prime numbers are also used in pseudorandom number generators including linear congruential generators and the Mersenne Twister.


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 ter ...
and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no-three-in-line problem, no three are in a line, or so that every triangle formed by three of the points Heilbronn triangle problem, has large area. Another example is Eisenstein's criterion, a test for whether a irreducible polynomial, 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 of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name. Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot (mathematics), knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The Prime decomposition (3-manifold), prime decomposition of 3-manifolds is another example of this type. Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas.


Constructible polygons and polygon partitions

Fermat primes are primes of the form :F_k = 2^+1, with k a nonnegative integer. 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 ...
, 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 polygon, regular n-gon is constructible polygon, 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 trisection, angle trisector if and only if the prime factors of regular polygon, n are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form 2^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 prime power, power of a prime number, but this is not known for other values of n.


Quantum mechanics

Beginning with the work of Hugh Lowell Montgomery, Hugh Montgomery and Freeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems. Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and SIC-POVM, symmetric informationally complete positive-operator-valued measures.


Biology

The evolutionary strategy used by cicadas of the genus ''Magicicada'' makes use of prime numbers. These insects spend most of their lives as larva, 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 plants are hypothesized to be smooth numbers, having only small prime numbers in their factorizations.


Arts and literature

Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as ''La Nativité du Seigneur'' (1935) and ''Quatre études de rythme'' (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations". In his science fiction novel ''Contact (novel), Contact'', scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975. In the novel ''The Curious Incident of the Dog in the Night-Time'' by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel ''The Solitude of Prime Numbers (novel), The Solitude of Prime Numbers'', in which they are portrayed as "outsiders" among integers.


Notes


References


External links

* * Caldwell, Chris, The Prime Pages 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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