A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 is either a prime itself or can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 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 n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and . Algorithms much more efficient than trial division have been devised to test the primality of large numbers. These include the Miller–Rabin primality test, which is fast but has a small probability of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits.
There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.
Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.
A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself.^{[1]} Natural numbers greater than 1 that are not prime are called composite.
For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, while 1, 4, and 6 are not prime. 1 is excluded as a prime number, for reasons explained below. 2 is a prime number, since the only natural numbers dividing it are 1 and 2. 3 is prime, as no numbers other than 1 and itself divide evenly into it. 4 is composite, since 2 is a number that divides evenly into it, in addition to 1 and itself. 5 is prime as only 1 and itself divide evenly into it. 6 is divisible by 2 and 3, therefore it is not prime.
No even number greater than 2 is prime because by definition, as any such even number n has at least three distinct divisors, namely 1, 2, and n. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 would end in 1, 3, 7, or 9, since even numbers are multiples of 2, and numbers ending in 0 or 5 are multiples of 5.
If n is a natural number, then 1 and n divide n without remainder. Therefore, the condition of being a prime can also be restated as: a number is prime if it is greater than one and if none of
divides n (without remainder). Yet another way to say the same is: a number n > 1 is prime if it cannot be written as a product of two integers a and b, both of which are larger than 1:
In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item.
The set of all primes is often denoted by P.
The first 168 prime numbers (all the prime numbers less than 1000) are:
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:
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.
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 mid-18th 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]}
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 zeta-function 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 APRT-CL, 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 public-key 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.
There are infinitely many prime numbers. Another way of saying this is that the sequence
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 (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 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.
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.
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.
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 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]}
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/(1-p)^{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 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 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 | |
Baillie-PSW 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 |
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 n − 1 or n + 1 is known. For example, the 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. These tests can be applied to check whether
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.
If only a partial factorization of n − 1 or n + 1 is known, there are extensions of these methods that may be able to prove that n is prime. ^{[21]}
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.^{[22]} The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.^{[23]} 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 256^{k} for some positive integer k, and searching for possible primes within the interval [256^{k}n + 1, 256^{k}(n + 1) − 1].
Type | Prime | Number of decimal digits | Date | Found by |
---|---|---|---|---|
Mersenne prime | 2^{74,207,281} − 1 | 22,338,618 | January 7, 2016^{[24]} | 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^{[25]} | Péter Szabolcs, PrimeGrid^{[26]} |
factorial prime | 208,003! − 1 | 1,015,843 | July 2016 | Sou Fukui^{[27]} |
primorial prime | 1,098,133# - 1 | 476,311 | March 2012 | James P. Burt, PrimeGrid^{[28]} |
twin primes | 2,996,863,034,895 × 2^{1,290,000} ± 1 | 388,342 | September 2016 | Tom Greer, PrimeGrid^{[29]} |
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.
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.^{[30]}
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 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 well understood.
There is no known efficient formula for primes. For example, there is no non-constant polynomial, even in several variables, that takes only prime values. However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on Wilson's theorem 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.^{[31]}
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
are prime for any natural number n in the first formula, and any number of exponents in the second formula.^{[32]} 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.^{[33]}
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) satisfies
which means 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 n-th 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).^{[34]} 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.
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,
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
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.^{[35]} 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).
Euler noted that the function
yields prime numbers for 0 ≤ n < 40,^{[36]}^{[37]} a fact leading into deep algebraic number theory: more specifically, Heegner numbers. 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 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 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).
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.^{[38]}^{[39]}
The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible. From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.
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, this conjecture has been verified for all numbers up to n = 2 · 10^{17}.^{[40]} 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.^{[41]} 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^{[42]} and infinitely many Mersenne primes, but not Fermat primes.^{[43]} 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.^{[44]} It is conjectured there are infinitely many primes of the form n^{2} + 1.^{[45]} 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.
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 British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.^{[46]} 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.
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 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, F_{5} is composite and so are all other Fermat numbers that have been verified as of 2015. A regular n-gon is constructible using straightedge and compass if and only if the odd prime factors of n (if any) are distinct Fermat primes.
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).
Several public-key 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.
The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers.^{[47]} 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.^{[48]} 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.^{[49]} 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.^{[50]}
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.^{[51]} 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.^{[52]} Prime models and prime 3-manifolds are other examples of this type.
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.
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.^{[53]}
Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring.^{[54]} 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 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
where x is an integer and p and q are (usual) prime numbers.^{[55]} 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.^{[56]}
Valuation theory studies certain functions from a field K to the real numbers R called valuations.^{[57]} 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 p-adic 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 p-adic norm is defined as ^{[58]}
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 p-adic norm |−|_{p} yields the field of p-adic numbers.^{[59]} 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 local-global principle again underlines the importance of primes to number theory.
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".^{[60]}
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.^{[61]} In the novel The Curious Incident of the Dog in the Night-Time by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers.^{[62]}
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.^{[63]} 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.^{[64]}
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