The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, 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 "factors.^{[2]} Primes can thus be considered the “basic building blocks” of the natural numbers. For example:

23244 = 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:
 n = p_{1} · p_{2} · ... · p_{t}
of a number n into (finitely many) prime factors p_{1}, p_{2}, ... to p_{t} is called prime factorization of n. The fundamental theorem of arithmetic can be rephrased so as to say that any factorization into primes will be identical except for the order of the factors. So, albeit there are many "prime factorization algorithms to do this in practice for larger numbers, they all have to yield the same result.
If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition is known as "Euclid's lemma.^{[3]} It is used in some proofs of the uniqueness of prime factorizations.
Primality of one[edit]
Most early Greeks did not even consider 1 to be a number,^{[4]} so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number.^{[5]} 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. ^{[6]} 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,^{[7]} started with 1 as its first prime.^{[8]} "Henri Lebesgue is said to be the last professional mathematician to call 1 prime.^{[9]} 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".^{[5]}
A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the "sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of "Euler's totient function or the "sum of divisors function.^{[10]}
History[edit]
There are hints in the surviving records of the "ancient Egyptians that they had some knowledge of prime numbers: the "Egyptian fraction expansions in the "Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the "Ancient Greeks. "Euclid'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 "perfect number from a "Mersenne prime. 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.
After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 "Pierre de Fermat stated (without proof) "Fermat's little theorem (later proved by "Leibniz and "Euler). Fermat also conjectured that all numbers of the form 2^{2n} + 1 are prime (they are called "Fermat numbers) and he verified this up to n = 4 (or 2^{16} + 1). However, the very next Fermat number 2^{32} + 1 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 2^{p} − 1, with p a prime. They are called Mersenne primes in his honor.
Euler's work in number theory included many results about primes. He showed the "infinite series "1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is "divergent. In 1747 he showed that the even "perfect numbers are precisely the integers of the form 2^{p−1}(2^{p} − 1), where the second factor is a Mersenne prime.
At the start of the 19th century, Legendre and Gauss independently conjectured that as x tends to infinity, the number of primes up to x is "asymptotic to x/ln(x), where ln(x) is the "natural logarithm of x. Ideas of Riemann in his "1859 paper on the zetafunction sketched a program that would lead to a proof of the prime number theorem. This outline was completed by "Hadamard and "de la Vallée Poussin, who independently proved the "prime number theorem in 1896.
Proving a number is prime is not done (for large numbers) by trial division. Many mathematicians have worked on "primality tests for large numbers, often 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),^{[11]} and the generalized "Lucas primality test. More recent algorithms like "APRTCL, "ECPP, and "AKS work on arbitrary numbers but remain much slower.
For a long time, prime numbers were thought to have extremely limited application outside of "pure mathematics.^{[12]} 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.
Since 1951 all the "largest known primes have been found by "computers. 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, while mathematicians continue to struggle with the theory of primes.
Number of prime numbers[edit]
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, "Goldbach's "proof based on "Fermat numbers,^{[13]} "Furstenberg's "proof using general topology,^{[14]} and "Kummer's elegant proof.^{[15]}
Euclid's proof[edit]
"Euclid's proof (Book IX, Proposition 20^{[16]}) considers any finite set S of primes. The key idea is to consider the "product of all these numbers plus one:
Like any other natural number, N is divisible by at least one prime number (it is possible that N itself is prime).
None of the primes by which N is divisible can be members of the finite set S of primes with which we started, because dividing N by any one of these leaves a remainder of 1. Therefore, the primes by which N is divisible are additional primes beyond the ones we started with. Thus any finite set of primes can be extended to a larger finite set of primes.
It is often erroneously reported that Euclid begins with the assumption that the set initially considered contains all prime numbers, leading to a "contradiction, or that it contains precisely the n smallest primes rather than any arbitrary finite set of primes.^{[17]} Today, the product of the smallest n primes plus 1 is conventionally called the nth "Euclid number.
Euler's analytical proof[edit]
"Euler's proof uses the "partial sums of the "reciprocals of primes,
For any arbitrary "real number x, there exists a prime p for which this partial sum is bigger than x.^{[18]} 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.^{[19]} For comparison, the sum
does not grow to infinity as n 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 "twin primes,
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[edit]
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[edit]
The most basic method of checking the primality of a given integer n is called "trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the "square root 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 is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. 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 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 is approximately given by , so the algorithm may need up to this number of trial divisions to check the primality of n. For n = 10^{20}, this number is 450 million—too large for many practical applications.
Sieves[edit]
An algorithm yielding all primes up to a given limit, such as required in the primesonly trial division method, is called a prime number "sieve. 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 10^{7} were also used.^{[20]}
Primality testing versus primality proving[edit]
Modern primality tests for general numbers n 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 p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1p)^{n}, 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 n^{p}≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out n^{b} (mod b) for a random value of n 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 "BailliePSW, "MillerRabin, and "SolovayStrassen tests, are guaranteed to 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 n, the number to be tested and, for probabilistic algorithms, the number k of tests performed. Moreover, ε 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 n, but on ε) times log^{5+ε}(n).
Test  Developed in  Type  Running time  Notes 

"AKS primality test  2002  deterministic  O(log^{6+ε}(n))  
"Elliptic curve primality proving  1977  deterministic  O(log^{5+ε}(n)) heuristically  
"BailliePSW primality test  1980  probabilistic  O(log^{3} n)  no known counterexamples 
"Miller–Rabin primality test  1980  probabilistic  O(k · log^{2+ε} (n))  error probability 4^{−k} 
"Solovay–Strassen primality test  1977  probabilistic  O(k · log^{3} n)  error probability 2^{−k} 
"Fermat primality test  probabilistic  O(k · log^{2+ε} (n))  fails for Carmichael numbers 
Specialpurpose algorithms and the largest known prime[edit]
In addition to the aforementioned tests applying to any natural number n, a number of much more efficient primality tests is available for special numbers. For example, to run "Lucas' primality test requires the knowledge of the prime factors of n − 1, while the "Lucas–Lehmer primality test needs the prime factors of n + 1 as input. For example, these tests can be applied to check whether
 "n! ± 1 = 1 · 2 · 3 · ... · n ± 1
are prime. Prime numbers of this form are known as "factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the "Sophie Germain primes (primes of the form 2p + 1 with p prime), "primorial primes, "Fermat primes and "Mersenne primes, that is, prime numbers that are of the form 2^{p} − 1, 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.
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.^{[21]} The "Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.^{[22]} 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 + 1, 256^{k}(n + 1) − 1].^{["citation needed]}
Type  Prime  Number of decimal digits  Date  Found by 

"Mersenne prime  2^{74,207,281} − 1  22,338,618  January 7, 2016^{[23]}  "Curtis Cooper, "Great Internet Mersenne Prime Search 
not a Mersenne prime ("Proth number)  10,223 × 2^{31,172,165} + 1  9,383,761  October 31, 2016^{[24]}  Péter Szabolcs, "PrimeGrid^{[25]} 
"factorial prime  208,003! − 1  1,015,843  July 2016  Sou Fukui^{[26]} 
"primorial prime  1,098,133#  1  476,311  March 2012  James P. Burt, "PrimeGrid^{[27]} 
"twin primes  2,996,863,034,895 × 2^{1,290,000} ± 1  388,342  September 2016  Tom Greer, "PrimeGrid^{[28]} 
Integer factorization[edit]
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 "elliptic curve.
Distribution[edit]
In 1975, number theorist "Don Zagier commented that primes both
grow like weeds among the natural numbers, seeming to obey no other law than that of chance [but also] exhibit stunning regularity [and] that there are laws governing their behavior, and that they obey these laws with almost military precision.^{[29]}
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 the consecutive integers from to (inclusive) are all composite (as is divisible by for between and ).
"Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
with "coprime 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 wellunderstood.
Formulas for primes[edit]
There is no known efficient formula for primes. For example, there is no nonconstant "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 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.^{[30]}
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
are prime for any natural number n in the first formula, and any number of exponents in the second formula.^{[31]} Here represents the "floor function, 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.^{[32]}
Number of prime numbers below a given number[edit]
The "prime counting function π(n) is defined as the number of primes not greater than n. For example, π(11) = 5, since there are five primes less than or equal to 11. There are known "algorithms to compute exact values of π(n) faster than it would be possible to compute each prime up to n. The prime number theorem states that π(n) is approximately given by
in the sense that the ratio of π(n) and the right hand fraction "approaches 1 when n grows to infinity. This implies that the likelihood that a number less than n is prime is (approximately) inversely proportional to the number of digits in n. A more accurate estimate for π(n) is given by the "offset logarithmic integral
The prime number theorem also implies "estimates for the size of the nth prime number p_{n} (i.e., p_{1} = 2, p_{2} = 3, etc.): up to a bounded factor, p_{n} grows like n log(n).^{[33]} In particular, the "prime gaps, i.e. the differences p_{n} − p_{n−1} of two consecutive primes, become arbitrarily large. This latter statement can also be seen in a more elementary way by noting that the sequence n! + 2, n! + 3, …, n! + n (for the notation n! read "factorial) consists of n − 1 composite numbers, for any natural number n. However, n − 1 composite numbers do make up gaps much smaller than n!. For example with n − 1 = 7, the first prime gap of 8 is between the primes 89 and 97 while 8! = 40320.
Arithmetic progressions[edit]
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 q = 9. Except for 3, none of these numbers is prime, since 3 + 9n = 3(1 + 3n) so that the remaining numbers in this progression are all composite. (In general terms, all prime numbers above q are of the form "q#·n + m, where 0 < m < q#, and m has no prime factor ≤ q.) Thus, the progression
 a, a + q, a + 2q, a + 3q, …
can have infinitely many primes only when a and q are "coprime, 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 picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.
The "Green–Tao theorem shows that there are arbitrarily long arithmetic progressions consisting of primes.^{[34]} An odd prime p is expressible as the sum of two squares, p = x^{2} + y^{2}, exactly if p is congruent 1 modulo 4 ("Fermat's theorem on sums of two squares).
Prime values of quadratic polynomials[edit]
Euler noted that the function
yields prime numbers for 0 ≤ n < 40,^{[35]}^{[36]} a fact leading into deep "algebraic number theory: more specifically, "Heegner numbers. For greater n, the expression also produces composite values. The "HardyLittlewood conjecture F makes an asymptotic prediction about the density of primes among the values of "quadratic polynomials (with integer "coefficients a, b, and c),
in terms of Li(n) and the coefficients a, b, and c. However, progress has been difficult. No quadratic polynomial (with a ≠ 0) is known to take infinitely many prime values. 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.
Open questions[edit]
Zeta function and the Riemann hypothesis[edit]
The "Riemann zeta function ζ(s) is defined as an "infinite sum
where s is a "complex number with "real part bigger than 1. It is a consequence of the fundamental theorem of arithmetic that this sum agrees with the "infinite product
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 1 + 1/2 + 1/3 + 1/4 + ... "diverges (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 ("Basel problem), due to Euler,
The reciprocal of ζ(2), 6/π^{2}, is the "probability that two numbers "selected at random are "relatively prime.^{[37]}^{[38]}
The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., 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.^{["clarification needed]} 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[edit]
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. As of February 2011^{[update]}, this conjecture has been verified for all numbers up to n = 2 · 10^{17}.^{[39]} Weaker statements than this have been proven, for example "Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. "Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a "semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes.^{[40]} 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 "Fibonacci primes^{[41]} and infinitely many "Mersenne primes, but not "Fermat primes.^{[42]} It is not known whether or not there are an infinite number of "Wieferich primes and of prime "Euclid numbers.
A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many "twin primes, 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 2n.^{[43]} It is conjectured there are infinitely many primes of the form n^{2} + 1.^{[44]} 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.
Applications[edit]
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 selfinterest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as "British mathematician "G. H. Hardy prided themselves on doing work that had absolutely no military significance.^{[45]} 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 "hash tables and "pseudorandom number generators.
Some "rotor machines 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.
Arithmetic modulo a prime and finite fields[edit]
Modular arithmetic modifies usual arithmetic by only using the numbers
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 "remainder 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
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 "addition and "multiplication familiar from the "integers remain valid in modular arithmetic. In the parlance of "abstract algebra, the above set of integers, which is also denoted Z/nZ, is therefore a "commutative ring for any n. "Division, however, is not in general possible in this setting. For example, for n = 6, the equation
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 2 ≤ m ≤ n − 1 are "coprime to n, i.e. their only "common divisor is one. Indeed, for n = 7, the equation
has a unique solution, x = 3. Because of this, for any prime p, Z/pZ (also denoted F_{p}) is called a "field or, more specifically, a "finite field 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
for any integer a not divisible by p, may be proved using these notions. This implies
"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 p − 1 or a divisor of p − 1. 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 "factorial (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[edit]
Fermat primes are primes of the form
 F_{k} = 2^{2k} + 1,
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, F_{5} is composite and so are all other Fermat numbers that have been verified as of 2015. A "regular ngon 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[edit]
Many mathematical domains make great use of prime numbers. An example from the theory of "finite groups are the "Sylow theorems: if G is a finite group and p^{n} is the "highest power of the prime p that divides the "order of G, then G has a subgroup of order p^{n}. Also, any group of prime order is cyclic ("Lagrange's theorem).
Publickey cryptography[edit]
Several publickey cryptography algorithms, such as "RSA and the "Diffie–Hellman key exchange, are based on large prime numbers (2048"bit primes are common). RSA relies on the assumption that it is much easier (i.e., 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, while the reverse operation the "discrete logarithm is thought to be a hard problem.
Prime numbers in nature[edit]
The evolutionary strategy used by "cicadas of the genus "Magicicada make use of prime numbers.^{[46]} 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. 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.^{[47]} If Magicicadas appeared at a nonprime 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 200year period, average predator populations during hypothetical outbreaks of 14 and 15year cicadas would be up to 2% higher than during outbreaks of 13 and 17year cicadas.^{[48]} Though small, this advantage appears to have been enough to drive natural selection in favour of a primenumbered lifecycle for these insects.
There is speculation^{["by whom?]} that the zeros of the "zeta function are connected to the energy levels of complex quantum systems.^{[49]}
Generalizations[edit]
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 "prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the "finite field with p elements, whence the name.^{[50]} Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in "knot theory, a "prime knot is a "knot that is indecomposable in the sense that it cannot be written as the "knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.^{[51]} "Prime models and "prime 3manifolds are other examples of this type.
Prime elements in rings[edit]
Prime numbers give rise to two more general concepts that apply to elements of any "commutative ring 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
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 "Gaussian integers 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 "Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z[i], 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 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.
Prime ideals[edit]
In "ring theory, the notion of number is generally replaced with that of "ideal. Prime ideals, which generalize prime elements in the sense that the "principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in "commutative algebra, "algebraic number theory and "algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the "Lasker–Noether theorem, which expresses every ideal in a "Noetherian "commutative ring as an intersection of "primary ideals, which are the appropriate generalizations of "prime powers.^{[52]}
Prime ideals are the points of algebrogeometric objects, via the notion of the "spectrum of a ring.^{[53]} "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. 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.^{[54]} Early attempts to prove "Fermat's Last Theorem climaxed when "Kummer introduced "regular primes, primes satisfying a certain requirement concerning the failure of unique factorization in the ring consisting of expressions
where a_{0}, ..., a_{p}_{−1} are integers and ζ is a complex number such that "ζ^{p} = 1.^{[55]}
Valuations[edit]
"Valuation theory studies certain functions from a field K to the real numbers R called "valuations.^{[56]} 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 "padic valuation of a rational number q is defined to be the integer v_{p}(q), such that
where both r and s are not divisible by p. For example, v_{3}(18/7) = 2. The padic norm is defined as^{[nb 1]}
In particular, this norm gets smaller when a number is multiplied by p, in sharp contrast to the usual "absolute value (also referred to as the "infinite prime). While "completing Q (roughly, filling the gaps) with respect to the absolute value yields the "field of "real numbers, completing with respect to the padic norm −_{p} yields the field of "padic numbers.^{[57]} These are essentially all possible ways to complete Q, by "Ostrowski's theorem. Certain arithmetic questions related to Q or more general "global fields may be transferred back and forth to the completed (or "local) fields. This "localglobal principle again underlines the importance of primes to number theory.
In games, arts, and literature[edit]
Prime numbers have influenced many artists and writers. The French "composer "Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as "La Nativité du Seigneur (1935) and "Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".^{[58]}
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.^{[59]} 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.^{[60]}
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.^{[61]} 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.^{[62]}
See also[edit]
 "Adleman–Pomerance–Rumely primality test
 "Bonse's inequality
 "Brun sieve
 "Burnside theorem
 "Chebotarev's density theorem
 "Chinese remainder theorem
 "Cullen number
 "Illegal prime
 "List of prime numbers
 "Multiplicative number theory
 "Number field sieve
 "Pépin's test
 "Practical number
 "Prime ktuple
 "Primon gas
 "Quadratic residuosity problem
 "Ramanujan prime
 "RSA number
 "Smooth number
 "Superprime
 "Woodall number
Notes[edit]
 ^ Some sources also put .
 ^ Dudley, Underwood (1978), Elementary number theory (2nd ed.), W. H. Freeman and Co., "ISBN "9780716700760, p. 10, section 2
 ^ Dudley 1978, Section 2, Theorem 2
 ^ Dudley 1978, Section 2, Lemma 5
 ^ See, for example, David E. Joyce's commentary on "Euclid's Elements, Book VII, definitions 1 and 2.
 ^ ^{a} ^{b} Why is the number one not prime? (from the Prime Pages' list of frequently asked questions) by Chris K. Caldwell.
 ^ See for instance: L. Euler. Commentarii academiae scientiarum Petropolitanae 9 (1737), 160188. Variae observationes circa series infinitas, Theorema 19, p.187.
 ^ Riesel 1994, p. 36
 ^ Conway & Guy 1996, pp. 129–130
 ^ Derbyshire, John (2003), "The Prime Number Theorem", Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Washington, D.C.: Joseph Henry Press, p. 33, "ISBN "9780309085496, "OCLC 249210614
 ^ ""Arguments for and against the primality of 1".
 ^ The Largest Known Prime by Year: A Brief History Prime Curios!: 17014…05727 (39digits)
 ^ For instance, Beiler writes that number theorist "Ernst Kummer loved his "ideal numbers, 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. Beiler, Albert H. (1966), Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover, p. 2, "ISBN "9780486210964. Katz, Shaul (2004), "Berlin roots—Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem", Science in Context, 17 (12): 199–234, "doi:10.1017/S0269889704000092, "MR 2089305.
 ^ Letter in "Latin from Goldbach to Euler, July 1730.
 ^ Furstenberg 1955
 ^ Ribenboim 2004, p. 4
 ^ James Williamson (translator and commentator), The Elements of Euclid, With Dissertations, "Clarendon Press, Oxford, 1782, page 63, English translation of Euclid's proof
 ^ Hardy, Michael; Woodgold, Catherine (2009). "Prime Simplicity". "Mathematical Intelligencer. 31 (4): 44–52. "doi:10.1007/s0028300990648.
 ^ "Apostol, Tom M. (1976), Introduction to Analytic Number Theory, Berlin, New York: "SpringerVerlag, "ISBN "9780387901633, Section 1.6, Theorem 1.13
 ^ Apostol 1976, Section 4.8, Theorem 4.12
 ^ (Lehmer 1909).
 ^ "Record 12MillionDigit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. October 14, 2009. Retrieved 20100104.
 ^ "EFF Cooperative Computing Awards". Electronic Frontier Foundation. Retrieved 20100104.
 ^ Cooper, Curtis. "Mersenne Prime Number discovery  2^{74207281}1 is Prime!". Great Internet Mersenne Prime Search. Mersenne Research, Inc. Retrieved 19 January 2016.
 ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). Retrieved 20170103.
 ^ Chris K. Caldwell. "The Top Twenty: Largest Known Primes". "The Prime Pages. Retrieved 20170103.
 ^ Chris K. Caldwell. "The Top Twenty: Factorial". "The Prime Pages. Retrieved 20170103.
 ^ Chris K. Caldwell. "The Top Twenty: Primorial". "The Prime Pages. Retrieved 20170103.
 ^ Chris K. Caldwell. "The Top Twenty: Twin Primes". "The Prime Pages. Retrieved 20170103.
 ^ Havil 2003, p. 171
 ^ "Matiyasevich, Yuri V. (1999), "Formulas for Prime Numbers", in "Tabachnikov, Serge, Kvant Selecta: Algebra and Analysis, II, "American Mathematical Society, pp. 13–24, "ISBN "9780821819159.
 ^ E. M. Wright (1951). "A primerepresenting function". "American Mathematical Monthly. 58 (9): 616–618. "doi:10.2307/2306356. "JSTOR 2306356.
 ^ Kvant Selecta: Algebra and Analysis, Volume 2, edited by Serge Tabachnikov, p. 15
 ^ (Tom M. Apostol 1976), Section 4.6, Theorem 4.7
 ^ ("Ben Green & "Terence Tao 2008).
 ^ Hua (2009), pp. 176–177"
 ^ See list of values, calculated by "Wolfram Alpha
 ^ Caldwell, Chris. "What is the probability that gcd(n,m)=1?". The "Prime Pages. Retrieved 20130906.
 ^ "C. S. Ogilvy & J. T. Anderson Excursions in Number Theory, pp. 29–35, Dover Publications Inc., 1988 "ISBN 0486257789
 ^ Tomás Oliveira e Silva (20110409). "Goldbach conjecture verification". Ieeta.pt. Retrieved 20110521.
 ^ "Ramaré, O. (1995), "On šnirel'man's constant", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 22 (4): 645–706, retrieved 20080822.
 ^ Caldwell, Chris, The Top Twenty: Lucas Number at The "Prime Pages.
 ^ E.g., see Guy 1981, problem A3, pp. 7–8
 ^ Tattersall, J.J. (2005), Elementary number theory in nine chapters, "Cambridge University Press, "ISBN "9780521850148, p. 112
 ^ "Weisstein, Eric W. "Landau's Problems". "MathWorld.
 ^ Hardy 1940 "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."
 ^ Goles, E.; Schulz, O.; Markus, M. (2001). "Prime number selection of cycles in a predatorprey model". "Complexity. 6 (4): 33–38. "doi:10.1002/cplx.1040.
 ^ Paulo R. A. Campos; Viviane M. de Oliveira; Ronaldo Giro & Douglas S. Galvão. (2004), "Emergence of Prime Numbers as the Result of Evolutionary Strategy", "Physical Review Letters, 93 (9): 098107, "arXiv:qbio/0406017, "Bibcode:2004PhRvL..93i8107C, "doi:10.1103/PhysRevLett.93.098107.
 ^ "Invasion of the Brood". "The Economist. May 6, 2004. Retrieved 20061126.
 ^ Ivars Peterson (June 28, 1999). "The Return of Zeta". MAA Online. Archived from the original on October 20, 2007. Retrieved 20080314.
 ^ "Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211, Berlin, New York: "SpringerVerlag, "ISBN "9780387953854, "MR 1878556, Section II.1, p. 90
 ^ Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.B Heidelberger Akad. Wiss. Math.Nat. Kl. 1949 (1949), 57–104.
 ^ Eisenbud 1995, section 3.3.
 ^ Shafarevich, Basic Algebraic Geometry volume 2 (Schemes and Complex Manifolds), p. 5, section V.1
 ^ Neukirch, Algebraic Number theory, p. 50, Section I.8
 ^ Neukirch, Algebraic Number theory, p. 38, Section I.7
 ^ Endler, Valuation Theory, p. 1
 ^ Gouvea: padic numbers: an introduction, Chapter 3, p. 43
 ^ Hill, ed. 1995
 ^ "Carl Pomerance, Prime Numbers and the Search for Extraterrestrial Intelligence, Retrieved on December 22, 2007
 ^ Mark Sarvas, Book Review: The Curious Incident of the Dog in the NightTime, at The Modern Word, Retrieved on March 30, 2012
 ^ The music of primes, "Marcus du Sautoy's selection of films featuring prime numbers.
 ^ "Introducing Paolo Giordano". Books Quarterly.^{["dead link]}
References[edit]
 "Apostol, Thomas M. (1976), Introduction to Analytic Number Theory, New York: Springer, "ISBN "0387901639
 "Conway, John Horton; "Guy, Richard K. (1996), The Book of Numbers, New York: Copernicus, "ISBN "9780387979939
 "Crandall, Richard; "Pomerance, Carl (2005), Prime Numbers: A Computational Perspective (2nd ed.), Berlin, New York: "SpringerVerlag, "ISBN "9780387252827
 "Derbyshire, John (2003), Prime obsession, Joseph Henry Press, Washington, DC, "ISBN "9780309085496, "MR 1968857
 "Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: SpringerVerlag, "ISBN "9780387942681, "MR 1322960
 Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: "AddisonWesley, "ISBN "0201019841
 "Furstenberg, Harry (1955), "On the infinitude of primes", "The American Mathematical Monthly, Mathematical Association of America, 62 (5): 353, "doi:10.2307/2307043, "JSTOR 2307043
 "Green, Ben; "Tao, Terence (2008), "The primes contain arbitrarily long arithmetic progressions", "Annals of Mathematics, 167 (2): 481–547, "arXiv:math.NT/0404188, "doi:10.4007/annals.2008.167.481
 "Gowers, Timothy (2002), Mathematics: A Very Short Introduction, "Oxford University Press, "ISBN "9780192853615
 "Guy, Richard K. (1981), Unsolved Problems in Number Theory, Berlin, New York: SpringerVerlag, "ISBN "9780387905938
 Havil, Julian (2003), Gamma: Exploring Euler's Constant, "Princeton University Press, "ISBN "9780691099835
 "Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, "Cambridge University Press, "ISBN "9780521092272
 "Hardy, Godfrey Harold (1940), "A Mathematician's Apology, Cambridge University Press, "ISBN "9780521427067
 "Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, "ISBN "9781114541016
 Hill, Peter Jensen, ed. (1995), The Messiaen companion, Portland, Or: Amadeus Press, "ISBN "9780931340956
 Hua, L. K. (2009), Additive Theory of Prime Numbers, Translations of Mathematical Monographs, 13, AMS Bookstore, "ISBN "9780821849422
 "Lehmer, D. H. (1909), Factor table for the first ten millions containing the smallest factor of every number not divisible by 2, 3, 5, or 7 between the limits 0 and 10017000, Washington, D.C.: Carnegie Institution of Washington
 McCoy, Neal H. (1968), Introduction To Modern Algebra, Revised Edition, Boston: "Allyn and Bacon, "LCCN 6815225
 Narkiewicz, Wladyslaw (2000), The development of prime number theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Berlin, New York: SpringerVerlag, "ISBN "9783540662891
 "Ribenboim, Paulo (2004), The little book of bigger primes, Berlin, New York: SpringerVerlag, "ISBN "9780387201696
 Riesel, Hans (1994), Prime numbers and computer methods for factorization, Basel, Switzerland: Birkhäuser, "ISBN "9780817637439
 Sabbagh, Karl (2003), The Riemann hypothesis, Farrar, Straus and Giroux, New York, "ISBN "9780374250072, "MR 1979664
 du Sautoy, Marcus (2003), The music of the primes, HarperCollins Publishers, "ISBN "9780066210704, "MR 2060134
External links[edit]
""  Look up prime number in Wiktionary, the free dictionary. 
""  Wikinews has related news: 
""  Wikimedia Commons has media related to Prime number. 
 Hazewinkel, Michiel, ed. (2001), "Prime number", "Encyclopedia of Mathematics, Springer, "ISBN "9781556080104
 Caldwell, Chris, The "Prime Pages at primes.utm.edu.
 Prime Numbers on "In Our Time at the "BBC.
 An Introduction to Analytic Number Theory, by Ilan Vardi and Cyril Banderier
 Plus teacher and student package: prime numbers from Plus, the free online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.
 A000040
Prime number generators and calculators[edit]
 Prime Number Checker identifies the smallest prime factor of a number.
 Fast Online primality test with factorization makes use of the Elliptic Curve Method (up to thousanddigits numbers, requires Java).
 Huge database of prime numbers
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