HOME

TheInfoList



OR:

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. 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 because of the fundamental theorem of arithmetic: 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 object, 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'' wi ...
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, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, 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 numbers. 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 ...
, proven at the end of the 19th century, which says that the probability of a randomly chosen large number being prime is inversely
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compare ...
to its number of digits, that is, to its logarithm. 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 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, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. In abstract algebra, 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 with ...
s.


Definition and examples

A natural number (1, 2, 3, 4, 5, 6, etc.) is called a ''prime number'' (or a ''prime'') if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite numbers. 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 divisors 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 divisors, 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 capital ''P'') or by \mathbb (a blackboard bold 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 purchased ...
, 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 each ...
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. Euclid'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 ...
'' (c. 300 BC) proves the
infinitude of primes Euclid's theorem is a fundamental statement in number theory that asserts that there are Infinite set, infinitely many prime number, prime numbers. It was first proved by Euclid in his work ''Euclid's Elements, Elements''. There are several proofs ...
and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime. 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 mai ...
mathematician Ibn al-Haytham (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, 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 brought the innovations from Islamic mathematics back to Europe. His book '' Liber Abaci'' (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root. In 1640 Pierre de Fermat 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 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 primes, 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 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 to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes \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. 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 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 ...
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 ...
. 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 als ...
, that certain arithmetic progressions 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 (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 that ...
. 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 Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...
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 and ...
and other distributed computing 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 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 (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps 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 in ...
,
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, 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 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 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 ...
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 ot ...
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".


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 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 suf ...
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, 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 ma ...
, 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 con ...
based on Fermat numbers, 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 number In mathematics, Euclid numbers are integers of the form , where ''p'n''# is the ''n''th primorial, i.e. the product of the first ''n'' prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theor ...
s. 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 In number theory, Mills' constant is defined as the smallest positive real number ''A'' such that the floor function of the double exponential function : \lfloor A^ \rfloor is a prime number for all natural numbers ''n''. This constant is named ...
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, 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 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 parity (mathematics), even number can be written as the sum of either two prime number, primes, or a prime and a semiprime (the product of two primes). History The theorem wa ...
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 gaps, the differences between consecutive primes. The existence of arbitrarily large prime gaps can be seen by noting that the sequence n!+2,n!+3,\dots,n!+n consists of n-1 composite numbers, for any natural number n. 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 primes, pairs of primes with difference 2; this is the twin prime conjecture.
Polignac's conjecture In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: :For any positive even number ''n'', there are infinitely many prime gaps of size ''n''. In other words: There are infinitely many cases of two consecutive ...
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 Andrica's conjecture (named afteDorin Andrica is a conjecture regarding the gaps between prime numbers. The conjecture states that the inequality :\sqrt - \sqrt < 1 holds for all n, where p_n is the ''n''th prime ...
,
Brocard's conjecture In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (''p'n'')2 and (''p'n''+1)2, where ''p'n'' is the ''n''th prime number, for every ''n'' ≥ 2. The conjecture is named after Hen ...
,, 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 mathemat ...
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, 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 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 Diric ...
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 value ...
s, limits, infinite series, 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 referr ...
. 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 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 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 ...
. 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 (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 ...
, 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 als ...
, 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 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). 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 primes, : \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 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 is t ...
\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 ...
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 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 als ...
asserts that the progression contains infinitely many primes. The Green–Tao theorem 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 ob ...
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 factoriza ...
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 c ...
. 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 polynomials 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 var ...
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 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 The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According t ...
, 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 ...
, 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 on the complex numbers. For complex numbers s with real part greater than one it equals both an 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 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 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 ...
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 In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied ...
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 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, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
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 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) \t ...
(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). Constructio ...
, 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 Valuation may refer to: Economics *Valuation (finance), the determination of the economic value of an asset or liability **Real estate appraisal, sometimes called ''property valuation'' (especially in British English), the appraisal of land or bui ...
(certain mappings from the multiplicative group of the field to a totally ordered additive group, also called orders),
absolute values 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), a ...
(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). Constructio ...
s 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 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 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 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 sp ...
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 of ...
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), 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 /math>, 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 Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
, 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,
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 ob ...
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, which expresses every ideal in a Noetherian
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 sp ...
as an intersection of primary ideals, which are the appropriate generalizations of prime powers. 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 the ...
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 ramification of prime ideals when lifted to an extension field, 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 deno ...
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 ...
s 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
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 * Christo ...
's introduction of regular primes, 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, 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 marked ...
s 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, any group of prime order is a cyclic group, 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 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 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 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''. 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
grows exponentially Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a q ...
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 (probability theory), 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 ...
(or 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 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, 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 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 argument A heuristic argument is an argument that reasons from the value of a method or principle that has been shown experimentally (especially through trial-and-error) to be useful or convincing in learning, discovery and problem-solving, but whose line ...
s rather than rigorous proofs. The AKS primality test 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 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 (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 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. 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 and ...
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 ci ...
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 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 considerab ...
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. 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 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 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) 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''b' ...
) 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 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 unti ...
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 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, 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 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 In the mathematical field of topology, knot theory is the study of knot (mathematics), 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 ...
, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The 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 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 n ...
. They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime, 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 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, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures.


Biology

The evolutionary strategy used by cicadas of the genus ''
Magicicada The term periodical cicada is commonly used to refer to any of the seven species of the genus ''Magicicada'' of eastern North America, the 13- and 17-year cicadas. They are called periodical because nearly all individuals in a local population a ...
'' 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 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 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 Defi ...
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 ''La Nativité du Seigneur'' (''The Nativity of the Lord'' or ''The Birth of the Saviour'') is a work for organ, written by the French composer Olivier Messiaen in 1935. ''La Nativité du Seigneur'' is a testament to Messiaen's Christian faith, ...
'' (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 ext ...
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 ''The Curious Incident of the Dog in the Night-Time'' is a 2003 mystery novel by British writer Mark Haddon. Its title refers to an observation by the fictional detective Sherlock Holmes (created by Arthur Conan Doyle) in the 1892 short story ...
'' 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 Wr ...
, 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 ''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 Ve ...
'', 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


{{Portal bar, Mathematics, Science, History of science, Arithmetic Integer sequences Articles containing proofs