In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101.^[1]

An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC.^[2] The number 1 is by definition not a prime number. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any nonzero natural number n can be factored into primes, written as a product of primes or powers of primes (including the empty product of factors for 1). Moreover, this factorization is unique except for a possible reordering of the factors.

The property of being prime is called primality. Verifying the primality of a given number n can be done by trial division, that is to say dividing n by all smaller numbers m, thereby checking whether n is a multiple of m, and therefore not prime but a composite. For big primes, increasingly sophisticated algorithms which are faster than that technique have been devised.

There is no known formula yielding all primes and no composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large can be modeled. The first result in that direction is the prime number theorem which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or the logarithm of n. This statement has been proved at the end of the 19th century. The unproven Riemann hypothesis dating from 1859 implies a refined statement concerning the distribution of primes.

Despite being intensely studied, many fundamental questions around prime numbers remain open. For example, Goldbach's conjecture which asserts that any even natural number bigger than two is the sum of two primes, or the twin prime conjecture which says that there are infinitely many twin primes (pairs of primes whose difference is two), have been unresolved for more than a century, notwithstanding the simplicity of their statements.

Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, notably the notion of prime ideals.

Primes are applied in several routines in information technology, such as public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Searching for big primes, often using distributed computing, has stimulated studying special types of primes, chiefly Mersenne primes whose primality is comparably quick to decide. As of 2009, the largest known prime has about 13 million decimal digits.^[3]

[ Prime numbers and the fundamental theorem of arithmetic

A natural number is called a prime, a prime number or just prime if it has exactly two distinct divisors. Otherwise it is called composite. Therefore, 1 is not prime, since it has only one divisor, namely 1. However, 2 and 3 are prime, since they have exactly two divisors, namely 1 and 2, and 1 and 3, respectively. Next, 4, is composite, since it has 3 divisors: 1, 2, and 4.

Using symbols, a number n > 1 is prime if it cannot be written as a product of two factors a and b, both of which are larger than 1:

n = a · b.

The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic which states that every positive integer larger than 1 can be written as a product of one or more primes in a way which is unique except possibly for the order of the prime factors. Primes can thus be considered the “basic building blocks” of the natural numbers. For example, we can write

23244 = 2 · 2 · 3 · 13 · 149 = 2² · 3 · 13 · 149. (2² denotes the square or second power of 2.)

As in this example, the same prime factor may occur multiple times. A decomposition

n = p₁ · p₂ · ... · p_t

of a number n into (finitely many) prime factors p₁, p₂, ... 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.

The set of all primes is often denoted P.

[ Examples and first properties

Illustration showing that 11 is a prime number while 12 is not.

The only even prime number is 2, since any larger even number is divisible by 2. Therefore, the term odd prime refers to any prime number greater than 2.

The image at the right shows a graphical way to show that 12 is not prime. More generally, all prime numbers except 2 and 5, written in the usual decimal system, end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form 6n − 1 or 6n + 1, because all other numbers are divisible by 2 or 3. Generalizing this, all prime numbers above q are of form q#·n + m, where 0 < m < q, and m has no prime factor ≤ q.

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. It is used in some proofs of the uniqueness of prime factorizations.

[ Primality of one

Until the 19th century, most mathematicians considered the number 1 a prime, the definition being just that a prime is divisible only by 1 and itself but not requiring a specific number of distinct divisors. There is still a large body of mathematical work that is valid despite labeling 1 a prime, such as the work of Stern and Zeisel. Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956,^[4] started with 1 as its first prime.^[5] Henri Lebesgue is said to be the last professional mathematician to call 1 prime.^{[citation needed]} The change in label occurred so that the fundamental theorem of arithmetic, as stated, is valid, i.e., “each number has a unique factorization into primes.”^[6][7] 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.^[8]

[ History

The Sieve of Eratosthenes is a simple algorithm for finding all prime numbers up to a specified integer. It is the predecessor to the modern Sieve of Atkin, which is faster but more complex. The Sieve of Eratosthenes was created in the 3rd century BC by Eratosthenes, an ancient Greek mathematician.

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). A special case of Fermat's theorem may have been known much earlier by the Chinese. Fermat conjectured that all numbers of the form 2²ⁿ + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 2¹⁶ + 1). However, the very next Fermat number 2³² + 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 which 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),^[9] 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;^{[citation needed]} 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 in the last ten to fifteen years, while mathematicians continue to struggle with the theory of primes.

[ The number of prime numbers

There are infinitely many prime numbers. The oldest known proof for this statement, sometimes referred to as Euclid's theorem, is due to the Greek mathematician Euclid. Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Consider any finite set of primes. Multiply all of them together and add 1 (see Euclid number). The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of 1. Because all non-prime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. Either way, there is at least one more prime that was not in the finite set we started with. This argument applies no matter what finite set we began with. So there are more primes than any given finite number. (Euclid, Elements: Book IX, Proposition 20)

This previous argument explains why the product P of finitely many primes plus 1 must be divisible by some prime (possibly itself) not among those finitely many primes.

The proof is sometimes phrased in a way that falsely leads some readers to think that P + 1 must itself be prime, and think that Euclid's proof says the prime product plus 1 is always prime. This confusion arises when the proof is presented as a proof by contradiction and P is assumed to be the product of the members of a finite set containing all primes. Then it is asserted that if P + 1 is not divisible by any members of that set, then it is not divisible by any primes and "is therefore itself prime" (quoting G. H. Hardy^[10]). This sometimes leads readers to conclude mistakenly that if P is the product of the first n primes then P + 1 is prime. That conclusion relies on a hypothesis later proved false, and so cannot be considered proved. The smallest counterexample with composite P + 1 is

(2 × 3 × 5 × 7 × 11 × 13) + 1 = 30,031 = 59 × 509 (both primes).

Many more proofs of the infinity of primes are known. Adding the reciprocals of all primes together results in a divergent infinite series:

The proof of that statement is due to Euler. More precisely, if S(x) denotes the sum of the reciprocals of all prime numbers p with p ≤ x, then

S(x) = ln ln x + O(1) for x → ∞.

Another proof based on Fermat numbers was given by Goldbach.^[11] Kummer's is particularly elegant^[12] and Harry Furstenberg provides one using general topology.^[13]

Not only are there infinitely many primes, Dirichlet's theorem on arithmetic progressions asserts that in every arithmetic progression a, a + q, a + 2q, a + 3q, … where the positive integers a and q are coprime, there are infinitely many primes.

[ Verifying primality

In order to use primes, verifying that a given number n is prime or not is of crucial interest. There are several ways to achieve this aim. A sieve is an algorithm that yields all primes up to a given limit. The oldest such sieve is the sieve of Eratosthenes (see above), useful for relatively small primes. The modern sieve of Atkin is more complicated, but faster when properly optimized. Before the advent of computers, lists of primes up to bounds like 10⁷ were also used.^[14]

In practice, one often wants to check whether a given number is prime, rather than generate a list of primes as the two mentioned sieve algorithms do. The most basic method to do this, known as trial division, works as follows: given a number n, one divides n by all numbers m less than or equal to the square root of that number. If any of the divisions come out as an integer, then the original number is not a prime. Otherwise, it is a prime. Actually it suffices to do these trial divisions for m prime, only. While an easy algorithm, it quickly becomes impractical for testing large integers because the number of possible factors grows too rapidly as the number-to-be-tested increases: According to the prime number theorem expounded below, the number of prime numbers less than n is near n / (ln (n) − 1). So, to check n for primality the largest prime factor needed is just less than , and so the number of such prime factor candidates would be close to . This increases ever more slowly with n, but, because there is interest in large values for n, the count is large also: for n = 10 ²⁰ it is 450 million.

Modern primality test algorithms can be divided into two main classes, deterministic and probabilistic (or "Monte Carlo") algorithms. Probabilistic algorithms may report a composite number as a prime, but certainly do not identify primes as composite numbers; deterministic algorithms on the other hand do not have the possibility of such erring. The interest of probabilistic algorithms lies in the fact that they are often quicker than deterministic ones; in addition for most such algorithms the probability of erroneously identifying a composite number as prime is known. They typically pick a random number a called a "witness" and check some formula involving the witness and the potential prime n. After several iterations, they declare n to be "definitely composite" or "probably prime". For example, Fermat's primality test relies on Fermat's little theorem (see above). Thus, if

a^{p − 1} (mod p)

is unequal to 1, p is definitely composite. However, p may be composite even if a^{p − 1} = 1 (mod p) for all witnesses a, namely when p is a Carmichael number. In general, composite numbers that will be declared probably prime no matter what witness is chosen are called pseudoprimes for the respective test. However, the most popular probabilistic tests do not suffer from this drawback. The following table compares some primality 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.

[ Special types of primes

Construction of a regular pentagon. 5 is a Fermat prime.

There are many particular types of primes, for example qualified by various formulae, or by considering its decimal digits. Primes of the form 2^p − 1, where p is a prime number, are known as Mersenne primes. Their importance lies in the fact that there are comparatively quick algorithms testing primality for Mersenne primes.

Primes of the form 2²ⁿ + 1 are known as Fermat primes; a regular n-gon is constructible using straightedge and compass if and only if

n = 2ⁱ · m

where m is a product of any number of distinct Fermat primes and i is any natural number, including zero. Only five Fermat primes are known: 3, 5, 17, 257, and 65,537. Prime numbers p where 2p + 1 is also prime are known as Sophie Germain primes. A prime p is called primorial or prime-factorial if it has the form

p = n# ± 1

for some number n, where n# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. It is not known whether there are infinitely many primorial or factorial primes.

[ Location of the largest known prime

Since the dawn of electronic computers the largest known prime has almost always been a Mersenne prime because there exists a particularly fast primality test for numbers of this form, the Lucas–Lehmer primality test. The following table gives the largest known primes of the mentioned types.

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^kn + 1, 256^k(n + 1) − 1].

The Electronic Frontier Foundation has offered a US$100,000 prize to the first discoverers of a prime with at least 10 million digits. The prize may be awarded to GIMPS and the UCLA mathematics department for discovering the 47th known Mersenne prime mentioned in the table.[1][2] They also offer $150,000 and $250,000 for 100 million digits and 1 billion digits, respectively.

[ Generating prime numbers

There is no known formula for primes which is more efficient at finding primes than the methods mentioned above.

There is 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.

There is no polynomial, even in several variables, that takes only prime values. However, there are polynomials in several variables, whose positive values (as the variables take all positive integer values) are exactly the primes (for an example, see formula for primes).

Another formula is based on Wilson's theorem mentioned above, and generates the number 2 many times and all other primes exactly once. There are other similar formulas which also produce primes.

[ Distribution

The Ulam spiral. Black pixels show prime numbers.

Given the fact that there is an infinity of primes, it is natural to seek for patterns or irregularities in the distribution of primes. The problem of modeling the distribution of prime numbers is a popular subject of investigation for number theorists. The occurrence of individual prime numbers among the natural numbers is (so far) unpredictable, even though there are laws (such as the prime number theorem and Bertrand's postulate) that govern their average distribution. Leonhard Euler commented

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate.^[18]

In a 1975 lecture, Don Zagier commented

There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.^[19]

Euler noted that the function

n² + n + 41

gives prime numbers for n < 40 (but not necessarily so for bigger n), a remarkable fact leading into deep algebraic number theory, more specifically Heegner numbers. The Ulam spiral depicts all natural numbers in a spiral-like way. Surprisingly, prime numbers cluster on certain diagonals and not others.

[ The number of prime numbers below a given number

A chart depicting π(n) (blue), n / ln (n) (green) and Li (n), the offset logarithmic integral (red).

The prime-counting function π(n) is defined as the number of primes up to 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. Values as large as π(10²⁰) can be calculated quickly and accurately with modern computers.

For larger values of n, beyond the reach of modern equipment, the prime number theorem provides an estimate: π(n) is approximately n/ln(n). In other words, as n gets very large, the likelihood that a number less than n is prime is inversely proportional to the number of digits in n. Even better estimates are known; see for example Prime number theorem#The prime-counting function in terms of the logarithmic integral.

If n is a positive integer greater than 1, then there is always a prime number p with n < p < 2n (Bertrand's postulate).

[ Gaps between primes

A sequence of consecutive integers none of which is prime constitutes a prime gap. There are arbitrarily long prime gaps: for any natural number n larger than 1, the sequence (for the notation n! read factorial)

n! + 2, n! + 3, …, n! + n

is a sequence of n − 1 consecutive composite integers, since

n! + m = m · (1 · 2 · … · (m − 1) · (m + 1) … n + 1)

is composite for any 2 ≤ m ≤ n. On the other hand, the gaps get arbitrarily small in proportion to the primes: the quotient

(p_{i + 1} − p_i) / p_i,

where p_i denotes the ith prime number (i.e., p₁ = 2, p₂ = 3, etc.), approaches zero as i approaches infinity.

[ Open questions

[ The Riemann hypothesis

To state the Riemann hypothesis, one of the oldest, yet, as of 2009, unproven mathematical conjectures, it is necessary to understand the Riemann zeta function (s is a complex number with real part bigger than 1)

The second equality is a consequence of the fundamental theorem of arithmetics, and shows that the zeta function is deeply connected with prime numbers. For example, the fact (see above) that there are infinitely many primes can be read off from the divergence of the harmonic series:

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,

Riemann's hypothesis is concerned with the zeroes of the ζ-function (i.e., s such that ζ(s) = 0). 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 thee distribution of primes only comes from random noise. FroSource: this wikipedia article, under CC-BY-SA.