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A prime number (or a prime) is a 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 . For example, 5 is prime because 1 and 5 are its only positive integer , 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 : any greater than 1 is either a prime itself or can be expressed as a product of primes that is unique 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 . 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 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 .

There are 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 that a given, randomly chosen number is prime is inversely proportional to its number of digits, or to the 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 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 public-key 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 , such as and .


Definition and examples
A (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive , 1 and the number itself.
(1978). 9780716700760, W. H. Freeman and Co..
Natural numbers greater than 1 that are not prime are called .

For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,

(2018). 9780764107689, Barron's Educational Series. .
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:

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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 .

No 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.

(1997). 9780387982892, Springer. .
Similarly, when written in the usual 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).

(1988). 9780080960197, Elsevier. .
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.
(1997). 9780198501053, Oxford University Press. .
In other words, is prime if items cannot be divided up into smaller equal-size groups of more than one item,
(2018). 9781136636622, Routledge. .
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 )

(2018). 9780387227382, Springer.
or by \mathbb{P} (a capital P).
(2018). 9781118243824, John Wiley & Sons. .


Unique factorization and the primality of one

Fundamental theorem of arithmetic
The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic,
(2018). 9780538737586, Cengage Learning. .
which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime .,
[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.
(2018). 9780060935580, Harper Collins. .
For example:

>
= 2 · 2 · 3 · 13 · 149
= 22 · 3 · 13 · 149. (22 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.

(2018). 9780191092435, Oxford University Press. .
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.

(1998). 9780191500503, Oxford University Press. .


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 this issue, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6.
(1981). 9789004065055, BRILL. .
so they could not consider it to be a prime. A few mathematicians from this time also considered the prime and composite numbers to be subdivisions of the odd numbers, so they also did not consider 2 to be prime. However, Plato, Aristotle, 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., pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with ; 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,

(1994). 9780817637439, Birkhäuser. .
started with 1 as its first prime.
(1996). 9780387979939, Copernicus.
has been said to be the last professional mathematician to call 1 prime,
(2018). 9780309085496, Joseph Henry Press.
but G. H. Hardy did so even later. By the early 20th century, mathematicians began to arrive at the consensus that 1 is not a prime number, but rather forms its own special category as a "unit".

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

(2018). 9780883855638, Mathematical Association of America. .


History
The sieve of Eratosthenes is a simple algorithm for finding all prime numbers up to a specified integer. It was created in the 3rd century BC by Eratosthenes, an ancient Greek|ancient Greece mathematician.
There are hints in the surviving records of the that they had some knowledge of prime numbers: the Egyptian fraction expansions in the , for instance, have quite different forms for primes and for composites., review in Mathematical Reviews of It has also been suggested that the records a list of prime numbers.
(2018). 9780674504431, Harvard University Press. .
However, the earliest surviving records of the explicit study of prime numbers come from Ancient Greek mathematics. 's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a from a .
(2018). 9781441960528, Springer. .
The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

Wilson's theorem, characterizing the prime numbers as the solutions to the equation , was found around 1000 AD by an Islamic mathematician, (Alhazen). Ibn al-Haytham 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 al-Banna' al-Marrakushi, 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. brought Islamic mathematics back to Europe. His book (1202) was the first to describe 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 )., 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.

(2018). 9780883855843, Mathematical Association of America. .
The French monk looked at primes of the form , with a prime. They are called in his honor.
(2018). 9780124211711, Academic Press. .
Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.
(2018). 9789814487528, World Scientific. .
Euler's work in number theory included many results about primes. He showed the is .
(2018). 9783540662891, Springer.
In 1747 he proved Ibn al-Haytham's conjecture (now the Euclid–Euler theorem) that the even 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 to , where is the natural logarithm of . Ideas of Riemann in his 1859 paper on the zeta-function sketched an outline for proving this. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by and de la Vallée Poussin, independently of each other, and the result is now known as the prime number theorem. Another important 19th-century 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),

(2018). 9783642181924, Springer. .
Proth's theorem (around 1878),
(2018). 9780201870732, Addison-Wesley.
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
(2018). 9781466561861, CRC Press. .
but are slower than the algorithms for specific number forms.
(1991). 9780883853153, Cambridge University Press. .
Since 1951 all the largest known primes have been found by .A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers. See e.g.
(2018). 9781107010833, Cambridge University Press. .
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 .For instance, Beiler writes that number theorist loved his , closely related to the primes, "because they had not soiled themselves with any practical applications", and Katz writes that , known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as that had already shown themselves to be useful.

(1966). 9780486210964, Dover. .
.
This changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem algorithm.
(2018). 9781498702690, CRC Press. .
Important recent developments in the theory of prime numbers include the Green–Tao theorem (2004) on long arithmetic progressions of prime numbers, and 's 2013 proof that there exist infinitely many of bounded size., pp. 18 and 47.


Number of prime numbers
There are 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 , 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 , Goldbach's proof based on , Letter in from Goldbach to Euler, July 1730. Furstenberg's proof using general topology, and elegant proof.
(2018). 9780387201696, Springer-Verlag. .


Euclid's proof
Euclid's proof that there are infinitely many primes (Book IX, Proposition 20James Williamson (translator and commentator), The Elements of Euclid, With Dissertations, , Oxford, 1782, page 63, English translation of Euclid's proof) shows equivalently that every finite of primes misses at least one prime. The key idea is to multiply together the numbers in and add one:
N = 1 + \prod_{p\in S} p.
Because the resulting number is greater than one, it has at least one prime number (possibly itself) in its prime factorization. But this prime cannot be in , because dividing by any one of the primes in leaves a remainder of 1. Therefore, does not contain all the primes.

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 .

(1991). 9780201529890, Addison-Wesley.


Euler's analytical proof
Euler's proof uses the of the reciprocals of primes,

S(p) = \frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p.

For any arbitrary , 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

\frac 1 {1^2} + \frac 1 {2^2} + \frac 1 {3^2} + \cdots + \frac 1 {n^2} = \sum_{i=1}^n \frac 1 {i^2}

does not grow to infinity as goes to infinity (see ). In this sense, prime numbers occur more often than squares of natural numbers.

(2018). 9780691120607, Princeton University Press. .
Brun's theorem states that the sum of the reciprocals of ,

\left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1} + \frac{1}} \right) + \cdots = \sum\limits_{ \begin{smallmatrix} p \text{ prime, } \\ p + 2 \text { prime} \end{smallmatrix}} {\left( {\frac{1}{p} + \frac{1}} \right)},
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.


Testing primality and integer factorization
There are various methods to determine whether a given number n is prime. The most basic routine, trial division, is of little practical use because of its slowness. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for particular numbers. Most such methods only tell whether n is prime or not. Routines also yielding one (or all) prime factors of n are called factorization algorithms.


Trial division
The most basic method of checking the primality of a given integer n is called . This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if n=a b is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most \sqrt{n}. For example, for n = 37 , the trial divisions are by None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to \sqrt{n} is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary ( m = 2, 3, and 5), given that 4 and 6 are composite.

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 \sqrt{n} is approximately given by \sqrt{n} / \ln(\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.


Sieves
An algorithm yielding all primes up to a given limit, such as required in the primes-only trial division method, is called a prime number . The oldest example, the sieve of Eratosthenes (see above), is still the most commonly used. The sieve of Atkin is another option. Before the advent of computers, lists of primes up to bounds like 107 were also used.


Primality testing versus primality proving
Modern primality tests for general numbers can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability if applied to a composite number. If we repeat the test times and pass every time, then the probability that our number is composite is , which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

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 time-consuming 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 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 test2002deterministicO((log n)6+ ε)
Elliptic curve primality proving1977deterministicO((log n)5+ ε) heuristically
Baillie–PSW primality test1980probabilisticO((log n)3)no known counterexamples
Miller–Rabin primality test1980probabilisticO( k · (log n)2+ ε)error probability 4k
Solovay–Strassen primality test1977probabilisticO( k · (log n)3)error probability 2k
Fermat primality test probabilisticO( k · (log n)2+ ε)fails for Carmichael numbers


Special-purpose algorithms and the largest known prime
In addition to the aforementioned tests that apply to any natural number n, there are very efficient (deterministic) primality tests if the complete factorization of either or is known. For example, the Lucas primality test requires the knowledge of the prime factors of . This test can be applied to check whether
± 1 = 1 · 2 · 3 · ... · n ± 1
are prime. Prime numbers of this form are known as . Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2 p + 1 with p prime), , and , that is, prime numbers that are of the form , where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers.

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 semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval 256 k n.

277,232,917 − 123,249,425December 26, 2017Jonathan Pace, Great Internet Mersenne Prime Search
not a Mersenne prime ()10,223 × 231,172,165 + 19,383,761October 31, 2016Péter Szabolcs,
208,003! − 11,015,843July 2016Sou Fukui
1,098,133# − 1476,311March 2012James P. Burt,
2,996,863,034,895  × 21,290,000 ± 1388,342September 2016Tom Greer,


Integer factorization
Given a composite integer n, the task of providing one (or all) prime factors is referred to as factorization of n. Elliptic curve factorization is an algorithm relying on arithmetic on an .


Distribution
In 1975, number theorist commented that primes both

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 n-th prime is known.

There are arbitrarily long sequences of consecutive non-primes, 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

p(n) = a + bn
with 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 such polynomials with the same b have approximately the same proportions of primes.

The corresponding question for quadratic polynomials is less well understood.


Formulas for primes
There is no known efficient formula for primes. For example, there is no non-constant , 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 and generates the number 2 many times and all other primes exactly once. There is also a set of Diophantine equations in 9 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.
(1999). 9780821819159, American Mathematical Society. .
.

Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants A > 1 and μ such that

\left \lfloor A^{3^n}\right \rfloor \text{ and } \left \lfloor 2^{\cdots^{2^{2^\mu}}} \right \rfloor
are prime for any natural number n in the first formula, and any number of exponents in the second formula. Here \lfloor \,\cdot\, \rfloor represents the , i.e., the largest integer not greater than the number in question. However, computing A or μ requires the knowledge of infinitely many primes to begin with. Kvant Selecta: Algebra and Analysis, Volume 2, edited by Serge Tabachnikov, p. 15


Number of prime numbers below a given number
[[File:PrimeNumberTheorem.svg|A chart depicting the prime counting function (blue), (green) and (red).
(red)|right|thumb|300px]] The prime counting function is defined as the number of primes not greater than . For example, , since there are five primes less than or equal to 11. There are known to compute exact values of faster than it would be possible to compute each prime up to . The prime number theorem states that satisfies
\pi(n) \sim \frac n {\ln n},
which means that the ratio of and the right hand fraction approaches 1 when grows to infinity. This implies that the likelihood that a number less than is prime is (approximately) inversely proportional to the number of digits in . A more accurate estimate for is given by the offset logarithmic integral
\operatorname{Li}(n) = \int_2^n \frac{dt}{\ln t}.

The prime number theorem also implies 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 .


Arithmetic progressions
An arithmetic progression is the set of natural numbers that give the same remainder when divided by some fixed number  q called modulus. For example,
3, 12, 21, 30, 39, ...,
is an arithmetic progression modulo . Except for 3, none of these numbers is prime, since so that the remaining numbers in this progression are all composite. (In general terms, all prime numbers above q are of the form · n +  m, where 0 <  m <  q#, and m has no prime factor ≤  q.) Thus, the progression
a,
can have infinitely many primes only when a and q are , i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The Green–Tao theorem shows that there are arbitrarily long arithmetic progressions consisting of primes. An odd prime p is expressible as the sum of two squares, , exactly if p is congruent 1 modulo 4 (Fermat's theorem on sums of two squares).


Prime values of quadratic polynomials
Euler noted that the function
n^2 + n + 41
yields prime numbers for ,
(1965). 9780821849422, American Mathematical Society.
See list of values, calculated by a fact leading into deep algebraic number theory: more specifically, . For greater n, the expression also produces composite values. The Hardy-Littlewood conjecture F makes an asymptotic prediction about the density of primes among the values of quadratic polynomials (with integer a, b, and c),
f(n) = a x^2 + bx + c ,
in terms of Li( n) and the coefficients a, b, and c. However, progress has been difficult. No quadratic polynomial (with ) is known to take infinitely many prime values.

The depicts all natural numbers in a spiral-like 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).


Open questions

Zeta function and the Riemann hypothesis

The Riemann zeta function ζ( s) is defined as an infinite sum

\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s},
where s is a with bigger than 1. It is a consequence of the fundamental theorem of arithmetic that this sum agrees with the
\prod_{p \text{ prime}} \frac 1 {1-p^{-s}}.
The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (), due to Euler,
\zeta(2) = \prod_p \frac{1}{1-p^{-2}}= \frac{\pi^2}{6}.
The reciprocal of ζ(2), 6/2, is the that two numbers selected at random are .C. S. Ogilvy & J. T. Anderson Excursions in Number Theory, pp. 29–35, Dover Publications Inc., 1988

The unproven Riemann hypothesis, dating from 1859, states that except for all zeroes of the ζ-function have 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
In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, 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 . Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a , the product of two primes. Also, any even integer can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.

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 . and infinitely many , but not .E.g., see

(1981). 9780387905938, Springer-Verlag.
It is not known whether or not there are an infinite number of and of prime .

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.

(2018). 9780521850148, Cambridge University Press. .
, p. 112
It is conjectured there are infinitely many primes of the form  n2 + 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 n2 and ( n + 1)2 for every positive integer  n. It is implied by the stronger Cramér's conjecture.


Applications
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 the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.
(2018). 9780521427067, Cambridge University Press.
However, this vision 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. Prime numbers are also used for and pseudorandom number generators.

Some were designed with a different number of pins on each rotor, with the number of pins on any one rotor either prime, or to the number of pins on any other rotor. This helped generate the of possible rotor positions before repeating any position.

The International Standard Book Numbers work with a , which exploits the fact that 11 is a prime.


Arithmetic modulo a prime and finite fields
Modular arithmetic modifies usual arithmetic by only using the numbers
\{0, 1, 2, \dots, n-1 \},
where n is a fixed natural number called modulus. Calculating sums, differences and products is done as usual, but whenever a negative number or a number greater than n − 1 occurs, it gets replaced by the after division by n. For instance, for n = 7, the sum 3 + 5 is 1 instead of 8, since 8 divided by 7 has remainder 1. This is referred to by saying "3 + 5 is congruent to 1 modulo 7" and is denoted
3 + 5 \equiv 1 \pmod 7.
Similarly, 6 + 1 ≡ 0 (mod 7), 2 − 5 ≡ 4 (mod 7), since −3 + 7 = 4, and 3 · 4 ≡ 5 (mod 7) as 12 has remainder 5. Standard properties of and familiar from the remain valid in modular arithmetic. In the parlance of , the above set of integers, which is also denoted Z/ n Z, is therefore a for any n. Division, however, is not in general possible in this setting. For example, for n = 6, the equation

3 \cdot x \equiv 2 \pmod 6,

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 to n, i.e. their only common divisor is one. Indeed, for n = 7, the equation

3 \cdot x \equiv 2 \pmod 7,
has a unique solution, . Because of this, for any prime p, Z/ p Z (also denoted F p) is called a field or, more specifically, a since it contains finitely many, namely p, elements.

A number of theorems can be derived from inspecting F p in this abstract way. For example, Fermat's little theorem, stating

a^{p-1} \equiv 1 \pmod p
for any integer a not divisible by p, may be proved using these notions. This implies
\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p.
Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/ p is always a recurring decimal, whose period is or a divisor of . The fraction 1/ p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of  q. Wilson's theorem says that an integer p > 1 is prime if and only if the ( p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if ( n − 1)! is divisible by  n.


Fermat primes and constructible polygons
Fermat primes are primes of the form
,
with k a natural number. They are named after Pierre de Fermat, who conjectured that all such numbers are prime. This was based on the evidence of the first five numbers in this series—3, 5, 17, 257, and 65,537—being prime. However, F5 is composite and so are all other Fermat numbers that have been verified as of 2015. A is constructible using straightedge and compass if and only if the odd prime factors of n (if any) are distinct Fermat primes.


Other mathematical occurrences of primes
Many mathematical domains make great use of prime numbers. An example from the theory of are the : if G is a finite group and pn is the the order of G, then G has a subgroup of order pn. Also, any group of prime order is cyclic (Lagrange's theorem)

and any group whose order is divisible by only two primes is (the ).


Public-key cryptography
Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (2048- primes are common). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed ) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.


Prime numbers in nature
The evolutionary strategy used by of the genus make use of prime numbers. These insects spend most of their lives as 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. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems.


Generalizations
The concept of 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 is the smallest subfield of a field F containing both 0 and 1. It is either Q or the with p elements, whence the name.
(2018). 9780387953854, .
, Section II.1, p. 90
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 , a is a knot that is indecomposable in the sense that it cannot be written as the of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57–104. and prime 3-manifolds are other examples of this type.


Prime elements in rings
Prime numbers give rise to two more general concepts that apply to elements of any R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is

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

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 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
In , the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a as an intersection of , which are the appropriate generalizations of .
(1995). 9780387942681, Springer-Verlag.

Prime ideals are the points of algebro-geometric 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 , a basic problem of algebraic number theory, bears some resemblance with . Such ramification questions occur even in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations

x^2 \equiv p \pmod q,

where x is an integer and p and q are (usual) prime numbers.

(1999). 9783540653998, Springer-Verlag.
Early attempts to prove Fermat's Last Theorem climaxed when introduced , primes satisfying a certain requirement concerning the failure of unique factorization in the ring consisting of expressions
a_0 + a_1 \zeta + \cdots + a_{p-1} \zeta^{p-1}\, ,
where a0, ..., ap−1 are integers and ζ is a complex number such that ., Section I.7, p. 38


Valuations
studies certain functions from a field K to the real numbers R called valuations.Endler, Valuation Theory, p. 1 Every such valuation yields a topology on K, and two valuations are called equivalent if they yield the same topology. A prime of K (sometimes called a place of K) is an equivalence class of valuations. For example, the of a rational number q is defined to be the integer vp( q), such that
q = p^{v_p(q)} \frac {r}{s},
where both r and s are not divisible by p. For example, The p-adic norm is defined as Some sources also put \left| q \right|_p := e^{-v_p(q)}. .
\left| q \right|_p := p^{-v_p(q)}.
In particular, this norm gets smaller when a number is multiplied by p, in sharp contrast to the usual (also referred to as the ). While completing Q (roughly, filling the gaps) with respect to the absolute value yields the field of real numbers, completing with respect to the p-adic norm |−| p yields the field of .Gouvea: p-adic numbers: an introduction, Chapter 3, p. 43 These are essentially all possible ways to complete Q, by Ostrowski's theorem. Certain arithmetic questions related to Q or more general may be transferred back and forth to the completed (or ) fields. This local-global principle again underlines the importance of primes to number theory.


In games, arts, and literature
Prime numbers have influenced many artists and writers. The French used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".
(1995). 9780931340956, Amadeus Press. .

In his science fiction novel Contact, scientist 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 in 1975., Prime Numbers and the Search for Extraterrestrial Intelligence, Retrieved on December 22, 2007 In the novel The Curious Incident of the Dog in the Night-Time by , the narrator arranges the sections of the story by consecutive prime numbers.Mark Sarvas, Book Review: The Curious Incident of the Dog in the Night-Time , 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 novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.


See also


External links


Prime number generators and calculators

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