Prime number

A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) 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.^[1]

An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC,^[2] although the density of prime numbers within natural numbers is 0. 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 different 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. The simplest trial division method tests whether n is a multiple of each integer m between 2 and . If n is a multiple of any of these integers then it is a composite number, and so not prime; if it is not a multiple of any of these integers then it is prime. As this method requires up to trial divisions, it is only suitable for relatively small values of n. More sophisticated algorithms, which are much more efficient than trial division, have been devised to test the primality of large numbers.

There is no known useful formula that yields all of the prime numbers 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 proven since 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, there remain some open questions around prime numbers which can be stated simply. 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 2010, the largest known prime number has about 13 million decimal digits.^[3]

Prime numbers and the fundamental theorem of arithmetic

A natural number is called a prime or a prime number if it has exactly two distinct natural number divisors. Natural numbers greater than 1 that are not prime are 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 integers 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

One of the primary reasons to exclude 1 from the set of prime numbers is the fundamental theorem of arithmetic, which says that every positive integer x can be uniquely written as a product of primes. When x is itself prime, this factorization has only one prime (x itself) in it, and when x = 1 the factorization is the empty product. But if 1 were admitted as a prime, then any integer could be factored in an infinite number of ways. For example, in this case the number 3 could be factored as 1^k · 3 = 3 for any integer k.

More generally, in unique factorization domains, every non-zero element is a unique product of prime elements and a unit. The factorization would not be unique if products of units were allowed.

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.^[6] 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.”^[7][8] 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.^[9]

History

The Sieve of Eratosthenes is a simple algorithm for finding all prime numbers up to a specified integer. It was created in the 3rd century BC by Eratosthenes, an ancient Greek mathematician. (Click to see animation.)

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),^[10] 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; 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 attributed 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^[11]). 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 infinitude 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.^[12] Kummer's is particularly elegant^[13] and Harry Furstenberg provides one using general topology.^[14]

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. The recent Green–Tao theorem shows that there are arbitrarily long progressions consisting of primes.^[15]

Verifying primality

To use primes requires verifying whether a given number n is prime or not. There are several ways to achieve this. 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.^[16]

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 2^2k + 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 offered a US$100,000 prize to the first discoverers of a prime with at least 10 million digits. On October 22, 2009, the prize was awarded to the Great Internet Mersenne Prime Search (GIMPS) for discovering the 45th known Mersenne prime, which is 2^43,112,609 − 1. The UCLA mathematics department owns the computer on which the discovery was made and received half of the prize money, with the remainder going to charity and future research.^[20] The EFF also offers $150,000 and $250,000 for 100 million digits and 1 billion digits, respectively.^[21]

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.^[22]

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.^[23]

Euler noted that the function

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.

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 · (n!/m + 1) = 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 2010, 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:

If there were a finite number of primes then would have a finite value - but instead we know that the Riemann zeta function has a simple pole at 1.

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 the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about 1/ 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

Besides the Riemann hypothesis, there are many more conjectures about prime numbers, many of which are old: for example, all four of Landau's problems from 1912 (the Goldbach, twin prime, Legendre conjecture and conjecture about n²+1 primes) are still unsolved.

Many 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^[24] and infinitely many Mersenne primes, but not Fermat primes.^[25] It is not known whether or not there are an infinite number of prime Euclid numbers.

A number of conjectures concern 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 which differ by 2n. It is conjectured there are infinitely many primes of the form n² + 1.^[26] These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n² and (n + 1)² for every positive integer n. It is implied by the stronger Cramér's conjecture.

Other conjectures relate the additive aspects of numbers with prime numbers: Goldbach's conjecture asserts that every even integer greater than 2 can be written as a sum of two primes, while the weak version states that every odd integer greater than 5 can be written as a sum of three primes.

Applications

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.^[27] 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 p

Modular arithmetic is a modification of usual arithmetic, by doing all calculations "modulo" a fixed number n. All calculations of modular arithmetic take place in the finite set

{0, 1, 2, ..., n − 1}.

Calculating modulo n means that sums, differences and products are calculated as usual, but then only the remainder after division by n is considered. For example, let n = 7. Then, in modular arithmetic modulo 7, the sum 3 + 5 is 1 instead of 8, since 8 divided by 7 has remainder 1. Similarly, 6 + 1 = 0 modulo 7, 2 − 5 = 4 modulo 7 (since −3 + 7 = 4) and 3 · 4 = 5 modulo 7 (12 has remainder 5). Standard properties of addition and multiplication familiar from the number system of the integers or rational numbers remain valid, for example

(a + b) · c = a · c + b · c (law of distributivity).

In general it is, however, not possible to divide in this setting. For example, for n = 6, the equation

3 · x = 2 (modulo 6),

a solution x of which would be an analogue of 2/3, cannot be solved, as one can see by calculating 3 · 0, ..., 3 · 5 modulo 6. The distinctive feature of prime numbers is the following: division is possible in modular arithmetic if and only if n is a prime. For n = 7, the equation

3 · x = 2 (modulo 7)

has a unique solution, x = 3. Equivalently, n is prime if and only if all integers m satisfying 2 ≤ m ≤ n − 1 are coprime to n, i.e., their greatest common divisor is 1. Using Euler's totient function φ, n is prime if and only if φ(n) = n − 1.

The set {0, 1, 2, ..., n − 1}, with addition and multiplication is denoted Z/nZ for all n. In the parlance of abstract algebra, it is a ring, for any n, but a finite field if and only if n is prime. A number of theorems can be derived from inspecting Z/pZ in an abstract way. For example Fermat's little theorem, stating that a^p − a is divisible by p for any integer a, may be proved using these notions. A consequence of this 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.

Other mathematical occurrences of 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 pⁿ is the highest power of the prime p which divides the order of G, then G has a subgroup of order pⁿ. Also, any group of prime order is cyclic (Lagrange's theorem).

Public-key cryptography

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example 512 bit primes are frequently used for RSA and 1024 bit primes are typical for Diffie–Hellman.). RSA relies on the fact that it is thought to be 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

Inevitably, some of the numbers that occur in nature are prime. There are, however, relatively few examples of numbers that appear in nature because they are prime.

One example of the use of prime numbers in nature is as an evolutionary strategy used by cicadas of the genus Magicicada.^[28] These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 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 specialise as predators on Magicicadas.^[29] 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.^[30] 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.^[31]

Generalizations

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the 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. 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 which 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.^[32] Prime models and prime 3-manifolds are other examples of this type.

Prime elements in rings

Prime numbers give rise to two more general concepts that apply to elements of any 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 if it is not a unit (i.e., does not have a multiplicative inverse) and the following property holds: given x and y in R such that p divides the product, then p divides at least one factor. Irreducible elements are ones which cannot be written as a product of two ring elements that are not units. In general, this is a weaker condition, but for any unique factorization domain, such as the ring Z of integers, the set of prime elements equals the set of irreducible elements, which for Z is :

{…, −11, −7, −5, −3, −2, 2, 3, 5, 7, 11, …}.

A common example is the Gaussian integers Z[i], that is, the set of complex numbers of the form a + bi with a and b in Z. This is an integral domain, 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. Gaussian primes can be used in proving quadratic reciprocity, while Eisenstein primes play a similar role for cubic reciprocity.

Prime ideals

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 any ideal in a Noetherian commutative ring as the intersection of primary ideals, which are the appropriate generalizations of prime powers.^[33]

Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring. 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.

Primes in valuation theory

In algebraic number theory, yet another generalization is used. A starting point for valuation theory is the p-adic valuations, where p is a prime number. It tells what highest power p divides a given number n. Using that, the p-adic norm is set up, which, in contrast to the usual absolute value, gets smaller when a number is multiplied by p. The completion of Q (the field of rational numbers) with respect to this norm leads to Q_p, the field of p-adic numbers, as opposed to R, the reals, which are the completion with respect to the usual absolute value. To highlight the connection to primes, the absolute value is often called the infinite prime. These are essentially all possible ways to complete Q, by Ostrowski's theorem.

In an arbitrary field K, one considers valuations on K, certain functions from K to the real numbers R. 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.

Arithmetic questions related to, global fields such as Q may, in certain cases, be transferred back and forth to the completed fields (known as local fields), a concept known as local-global principle. This again underlines the importance of primes to number theory.

In the arts and literature

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 one of the études. According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".^[34]

In his science fiction novel Contact, later made into a film of the same name, the 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.^[35]

Many films reflect a popular fascination with the mysteries of prime numbers and cryptography: films such as Cube, Sneakers, The Mirror Has Two Faces and A Beautiful Mind, the latter of which is based on the biography of the mathematician and Nobel laureate John Forbes Nash by Sylvia Nasar.^[36] 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.^[37]

See also

Distributed computing projects that search for primes

• PrimeGrid searches for megaprimes.
• Wieferich@Home searches for Wieferich primes.
• GIMPS searches for Mersenne primes.

Notes

References

• Apostol, Thomas M. (1976), Introduction to Analytic Number Theory, New York: Springer, ISBN 0-387-90163-9 
• Conway, John Horton; Guy, Richard K. (1996), The Book of Numbers, New York: Copernicus, ISBN 978-0-387-97993-9 
• Crandall, Richard; Pomerance, Carl (2005), Prime Numbers: A Computational Perspective (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-25282-7 
• Derbyshire, John (2003), Prime obsession, Joseph Henry Press, Washington, DC, MR1968857, ISBN 978-0-309-08549-6 
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1 
• Furstenberg, Harry (1955), "On the infinitude of primes", The American Mathematical Monthly (Mathematical Association of America) 62 (5): 353, doi:10.2307/2307043, ISSN 0002-9890, http://www.jstor.org/stable/2307043 
• Green, Ben; Tao, Terence (2008), "The primes contain arbitrarily long arithmetic progressions", Annals of Mathematics 167: 481–547, doi:10.4007/annals.2008.167.481, arXiv:math.NT/0404188 
• Gowers, Timothy (2002), Mathematics: A Very Short Introduction, Oxford University Press, ISBN 978-0-19-285361-5 
• Guy, Richard K. (1981), Unsolved Problems in Number Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90593-8 
• Havil, Julian (2003), Gamma: Exploring Euler's Constant, Princeton University Press, ISBN 978-0-691-09983-5 
• Hardy, Godfrey Harold (1908), A Course of Pure Mathematics, Cambridge University Press, ISBN 978-0-521-09227-2 
• Hardy, Godfrey Harold (1940), A Mathematician's Apology, Cambridge University Press, ISBN 978-0-521-42706-7 
• 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 
• Narkiewicz, Wladyslaw (2000), The development of prime number theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66289-1 
• Ribenboim, Paulo (2004), The little book of bigger primes, Berlin, New York: Springer-Verlag, ISBN 978-0-387-20169-6 
• Riesel, Hans (1994), Prime numbers and computer methods for factorization, Basel, Switzerland: Birkhäuser, ISBN 978-0-8176-3743-9 
• Sabbagh, Karl (2003), The Riemann hypothesis, Farrar, Straus and Giroux, New York, MR1979664, ISBN 978-0-374-25007-2 
• du Sautoy, Marcus (2003), The Music of Primes website The music of the primes, HarperCollins Publishers, MR2060134, ISBN 978-0-06-621070-4, http://www.musicoftheprimes.com/ The Music of Primes website 

Further references

• Hill, Peter Jensen, ed. (1995), The Messiaen companion, Portland, Or: Amadeus Press, ISBN 978-0-931340-95-6 
• Kelly, Katherine E., ed. (2001), The Cambridge companion to Tom Stoppard, Cambridge University Press, ISBN 978-0-521-64592-8 
• Stoppard, Tom (1993), Arcadia, London: Faber and Faber, ISBN 978-0-571-16934-4 

External links

• Caldwell, Chris, The Prime Pages at primes.utm.edu.
• Prime Numbers at MathWorld
• MacTutor history of prime numbers
• The prime puzzles
• An English translation of Euclid's proof that there are infinitely many primes
• Number Spiral with prime patterns
• An Introduction to Analytic Number Theory, by Ilan Vardi and Cyril Banderier
• EFF Cooperative Computing Awards
• Why a Number Is Prime by Enrique Zeleny, Wolfram Demonstrations Project.
• 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
• Prime Numbers Spirals and Visual Patterns by Luis Mateos

Prime number generators and calculators

• Online Prime Number Generator and Checker - instantly checks and finds prime numbers up to 128 digits long (does NOT require Java or JavaScript)
• Prime Number Checker identifies the smallest prime factor of a number
• Fast Online primality test — Dario Alpern's personal site – Makes use of the Elliptic Curve Method (up to thousands digits numbers check!, requires Java)
• Prime Number Generator — Generates a given number of primes above a given start number.
• Primes from WIMS is an online prime generator.
• Huge database of prime numbers
• All prime numbers below 10,000,000,000

Divisibility-based sets of integers Overview Integer factorization · Divisor · Unitary divisor · Divisor function · Prime factor · Fundamental theorem of arithmetic Forms of factorization Prime number · Composite number · Semiprime number · Pronic number · Sphenic number · Square-free number · Powerful number · Perfect power · Achilles number · Smooth number · Regular number · Rough number · Unusual number Constrained divisor sums Perfect number · Almost perfect number · Quasiperfect number · Multiply perfect number · Hyperperfect number · Superperfect number · Unitary perfect number · Semiperfect number · Primitive semiperfect number · Practical number Numbers with many divisors Abundant number · Primitive abundant number · Highly abundant number · Superabundant number · Colossally abundant number · Highly composite number · Superior highly composite number · Weird number Numbers related to aliquot sequences Untouchable number · Amicable number · Sociable number Other Deficient number · Friendly number · Solitary number · Sublime number · Harmonic divisor number · Frugal number · Equidigital number · Extravagant number