A prime number (or a prime) is a natural number greater than 1 that has no positive other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer Factorization, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 is either a prime itself or can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, , , etc. are all valid factorizations of 3.
The property of being prime is called primality. A simple but slow method of verifying the primality of a given number is known as trial division. It consists of testing whether is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as . , the largest known prime number has 23,249,425 numerical digit.
There are Infinite set primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number is prime is inversely proportional to its number of digits, or to the logarithm of .
Many questions regarding prime numbers remain open, such as Goldbach's conjecture (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 whose difference is 2). 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 publickey cryptography, which makes use of properties such as the difficulty of factoring large numbers into their . Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as and .
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. 1 is not a prime number because it has only one divisor, itself. 4 has three divisors: 1, 2, and 4. And 6 has four divisors: 1, 2, 3, and 6. So none of 1, 4, or 6 are prime. 4 and 6 are both composite.The first 168 prime numbers (all the prime numbers less than 1000) are:
No even number greater than 2 is prime because each has at least three divisors: 1, 2, and itself. Therefore, the prime numbers other than two are all . They are often called odd primes.
Similarly, when written in the usual decimal system, all prime numbers larger than 5 must 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.Every natural number has both 1 and itself as a divisor. Therefore, if it has any other divisor, it cannot be prime. Based on this idea, the primes can be defined as the numbers without other divisors: a number is prime if it is greater than one and if none of the numbers divides evenly (without remainder).
Yet another way to say the same thing is that a number is prime if it cannot be written as a product of of two integers and , both of which are larger than 1. In other words, is prime if items cannot be divided up into smaller equalsize groups of more than one item, or if it is not possible to arrange dots into a rectangular grid that is more than one dot wide and more than one dot high.The set of all primes is often denoted by (a boldface capital P)
or by $\backslash mathbb\{P\}$ (a blackboard bold capital P).
[https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA16 Section 2, Theorem 2, p. 16] Primes can thus be considered the "basic building blocks" of the natural numbers. For example:
= 2 · 2 · 3 · 13 · 149 
= 2^{2} · 3 · 13 · 149. (2^{2} denotes the square or second power of 2.) 
As in this example, the same prime factor may occur multiple times. A decomposition:
of a number into (finitely many) prime factors , , ... to is called prime factorization of . The fundamental theorem of arithmetic can be rephrased so as to say that any two factorizations of the same number into primes will be identical except for the order of the factors.
So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result.If is a prime number and divides a product of integers, then divides or divides ., Section 2, Lemma 5, p. 15 This proposition is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations.
By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number., pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet. In the mid18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be a prime number., p. 15. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956,
started with 1 as its first prime.If the definition of a prime number were changed to call 1 a prime, many statements involving the prime numbers would not hold in the form they are usually stated, but would instead require special treatment for the number 1. For example, the number 15 can be factored as and , so if 1 were defined to be prime, the number 15 would have two different factorizations into prime numbers. In order for the fundamental theorem of arithmetic to remain valid with this definition, it would need to be rephrased in terms of factorizations into primes greater than 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 produce as 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 are different for prime numbers than they are for 1.For the totient, see , p. 245. For the sum of divisors, see
Wilson's theorem, characterizing the prime numbers as the solutions to the equation , was found around 1000 AD by an Islamic mathematician, Ibn alHaytham (Alhazen). Ibn alHaytham also investigated the perfect numbers formed from Mersenne primes, and conjectured that all perfect numbers arose in this way, but was unable to prove it. Another Islamic mathematician, Ibn alBanna' alMarrakushi, observed that the sieve of Eratosthenes can be sped up by testing only the divisors up to the square root of the largest number to be tested. Fibonacci brought 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.
The next significant developments took place in 17th and 18th century Europe. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Leonhard Euler)., 8. Fermat's Little Theorem (November 2003), p. 45 Fermat also conjectured that all numbers of the form are prime (they are called ) and he verified this up to (or ). However, the very next Fermat number is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime.
The French monk Marin Mersenne looked at primes of the form , with a prime. They are called Mersenne primes in his honor. Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler's work in number theory included many results about primes. He showed the infinite sum is Divergent series. In 1747 he proved Ibn alHaytham's conjecture (now the Euclid–Euler theorem) that the even perfect numbers are precisely the integers of the form , where the second factor is a Mersenne prime.At the start of the 19th century, Legendre and Gauss independently conjectured that as tends to infinity, the number of primes up to is asymptotic to , where is the natural logarithm of . Ideas of Riemann in his 1859 paper on the zetafunction sketched an outline for proving this. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Jacques Hadamard and de la Vallée Poussin, independently of each other, and the result is now known as the prime number theorem. Another important 19thcentury result was Dirichlet's theorem on arithmetic progressions, that every arithmetic progression contains infinitely many primes.
Many mathematicians have worked on for larger numbers than would be possible by trial division. Some of these methods are restricted to specific number forms; this includes Pépin's test for Fermat numbers (1877),
Proth's theorem (around 1878), the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test. More recent algorithms like the Adleman–Pomerance–Rumely primality test, Elliptic curve primality proving, and the AKS primality test work on arbitrary numbers but are slower than the algorithms for specific number forms. Since 1951 all the largest known primes have been found by .A 44digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers. See e.g. The search for ever larger primes has generated interest outside mathematical circles. The Great Internet Mersenne Prime Search and other distributed computing projects to find large primes have become popular,, p. 245. while mathematicians continue to struggle with the theory of primes.For a long time, prime numbers were thought to have extremely limited application outside of pure mathematics.For instance, Beiler writes that number theorist Ernst Kummer loved his , closely related to the primes, "because they had not soiled themselves with any practical applications", and Katz writes that Edmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as geometry that had already shown themselves to be useful.
. This changed in the 1970s when the concepts of publickey cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem algorithm. Important recent developments in the theory of prime numbers include the Green–Tao theorem (2004) on long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many of bounded size., pp. 18 and 47.
It is often erroneously reported that Euclid begins with the assumption that the set contains all prime numbers, leading to a contradiction, or that it contains the smallest primes up to some threshold rather than any arbitrary finite set of primes. The numbers formed as products of the smallest primes plus one are called the .
For any arbitrary real number , there exists a prime for which this partial sum is bigger than ., Section 1.6, Theorem 1.13 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 being unbounded. More precisely, the growth rate of S( p) is doubly logarithmic, as quantified by Mertens' second theorem., Section 4.8, Theorem 4.12 For comparison, the sum
does not grow to infinity as goes to infinity (see Basel problem). In this sense, prime numbers occur more often than squares of natural numbers.
Brun's theorem states that the sum of the reciprocals of ,
While a simple method, trial division quickly becomes impractical for testing large integers because the number of possible factors grows too rapidly as n increases. According to the prime number theorem explained below, the number of prime numbers less than $\backslash sqrt\{n\}$ is approximately given by $\backslash sqrt\{n\}\; /\; \backslash ln(\backslash sqrt\{n\})$, so the algorithm may need up to this number of trial divisions to check the primality of n. For , this number is 450 million—too large for many practical applications.
A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that for any if is a prime number. If we have a number that we want to test for primality, then we work out for a random value of as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie–PSW and Miller–Rabin tests, may fail at least some of the time when applied to a composite number.
Deterministic algorithms do not erroneously report composite numbers as prime. In practice, the fastest such method is known as elliptic curve primality proving. Analyzing its run time is based on heuristic arguments, as opposed to the rigorously proven complexity of the more recent AKS primality test. Deterministic methods are typically slower than probabilistic ones, so the latter ones are typically applied first before a more timeconsuming deterministic routine is employed.
The following table lists a number of prime tests. The running time is given in terms of , the number to be tested and, for probabilistic algorithms, the number of tests performed. Moreover, is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that, for example, elliptic curve primality proving requires a time that is bounded by a factor (not depending on , but on ) times .
AKS primality test  2002  deterministic  O((log n)^{6+ ε})  
Elliptic curve primality proving  1977  deterministic  O((log n)^{5+ ε}) heuristically  
Baillie–PSW primality test  1980  probabilistic  O((log n)^{3})  no known counterexamples 
Miller–Rabin primality test  1980  probabilistic  O( k · (log n)^{2+ ε})  error probability 4^{− k} 
Solovay–Strassen primality test  1977  probabilistic  O( k · (log n)^{3})  error probability 2^{− k} 
Fermat primality test  probabilistic  O( k · (log n)^{2+ ε})  fails for Carmichael numbers 
If only a partial factorization of or is known, there are extensions of these methods that may be able to prove that n is prime.
The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search 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. Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semirandom binary data, converting it to a number n, multiplying it by 256^{k} for some positive integer k, and searching for possible primes within the interval 256^{ k} n.
Mersenne prime  2^{77,232,917} − 1  23,249,425  December 26, 2017  Jonathan Pace, Great Internet Mersenne Prime Search 
not a Mersenne prime (Proth number)  10,223 × 2^{31,172,165} + 1  9,383,761  October 31, 2016  Péter Szabolcs, PrimeGrid 
factorial prime  208,003! − 1  1,015,843  July 2016  Sou Fukui 
primorial prime  1,098,133# − 1  476,311  March 2012  James P. Burt, PrimeGrid 
2,996,863,034,895 × 2^{1,290,000} ± 1  388,342  September 2016  Tom Greer, PrimeGrid 
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, but no efficient formula for the nth prime is known.
There are arbitrarily long sequences of consecutive nonprimes, as for every positive integer $n$ the $n$ consecutive integers from $(n+1)!\; +\; 2$ to $(n+1)!\; +\; n\; +\; 1$ (inclusive) are all composite (as $(n+1)!\; +\; k$ is divisible by $k$ for $k$ between $2$ and $n\; +\; 1$).
Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
The corresponding question for quadratic polynomials is less well understood.
Other examples of primegenerating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants A > 1 and μ such that
The prime number theorem also implies Approximation for the size of the th prime number (i.e., , , etc.): up to a bounded factor, grows like ., Section 4.6, Theorem 4.7 In particular, the , i.e. the differences of two consecutive primes become arbitrarily large. This latter statement can also be seen in a more elementary way by noting that the sequence consists of composite numbers, for any natural number . However, composite numbers do make up gaps much smaller than . For example, with , the first prime gap of 8 is between the primes 89 and 97 while .
The Ulam spiral depicts all natural numbers in a spirallike way. Primes cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others. The example below displays concentrations of prime numbers (blue background) from 41 to 1001 arranged in an Ulam spiral with a start value of 41 (green background numbers are numbers with just 3 divisors, and red background are numbers with a large number of divisors).
The Riemann zeta function ζ( s) is defined as an infinite sum
The unproven Riemann hypothesis, dating from 1859, states that except for all zeroes of the ζfunction have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible. From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.
Other conjectures deal with the question whether an infinity of prime numbers subject to certain constraints exists. It is conjectured that there are infinitely many Caldwell, Chris, The Top Twenty: Lucas Number at The Prime Pages. and infinitely many , but not .E.g., see
A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many , pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2 n.
, p. 112 It is conjectured there are infinitely many primes of the form n^{2} + 1. These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n^{2} and ( n + 1)^{2} for every positive integer n. It is implied by the stronger Cramér's conjecture.
Some were designed with a different number of pins on each rotor, with the number of pins on any one rotor either prime, or coprime to the number of pins on any other rotor. This helped generate the full cycle of possible rotor positions before repeating any position.
The International Standard Book Numbers work with a check digit, which exploits the fact that 11 is a prime.
a solution x of which would be an analogue of 2/3, cannot be solved, as one can see by calculating 3 · 0, ..., 3 · 5 modulo 6. The distinctive feature of prime numbers is the following: division is possible in modular arithmetic if and only if n is a prime. Equivalently, n is prime if and only if all integers m satisfying are coprime to n, i.e. their only common divisor is one. Indeed, for n = 7, the equation
A number of theorems can be derived from inspecting F_{ p} in this abstract way. For example, Fermat's little theorem, stating
and any group whose order is divisible by only two primes is solvable group (the Burnside theorem).
There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems.
In any ring R, any prime element is irreducible. The converse does not hold in general, but does hold for unique factorization domains.
The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Z i , that is, the set 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 . Not every prime (in Z ) is a Gaussian prime: in the bigger ring Zi, 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4 k + 3 are Gaussian primes, whereas rational primes of the form 4 k + 1 are not.
Prime ideals are the points of algebrogeometric objects, via the notion of the spectrum of a ring.Shafarevich, Basic Algebraic Geometry volume 2 (Schemes and Complex Manifolds), p. 5, section V.1 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 field extension, a basic problem of algebraic number theory, bears some resemblance with ramified cover. Such ramification questions occur even in numbertheoretic 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 solvability of quadratic equations
where x is an integer and p and q are (usual) prime numbers.
Early attempts to prove Fermat's Last Theorem climaxed when Ernst Kummer introduced , primes satisfying a certain requirement concerning the failure of unique factorization in the ring consisting of expressions
In his science fiction novel Contact, NASA scientist Carl Sagan suggested that prime numbers could be used as a means of communicating with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975.Carl Pomerance, Prime Numbers and the Search for Extraterrestrial Intelligence, Retrieved on December 22, 2007 In the novel The Curious Incident of the Dog in the NightTime by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers.Mark Sarvas, Book Review: The Curious Incident of the Dog in the NightTime , at The Modern Word , Retrieved on March 30, 2012
Many films, such as Cube, Sneakers, The Mirror Has Two Faces and A Beautiful Mind reflect a popular fascination with the mysteries of prime numbers and cryptography. Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.
