Prime number

 Divisibility-based sets of integers Form of factorization: Prime number Composite number Powerful number Square-free number Achilles number Constrained divisor sums: Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hyperperfect number Unitary perfect number Semiperfect number Primitive semiperfect number Practical number Numbers with many divisors: Abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Other: Deficient number Weird number Amicable number Sociable number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number See also: Divisor function Divisor Prime factor Factorization

In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. There exists an infinitude of prime numbers, as demonstrated by Euclid, in about 300 B.C.. The first 30 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, 103, 107, 109, and 113 (sequence A000040 in OEIS); see the list of prime numbers for a longer list.

The property of being a prime is called primality, and the word prime is also used as an adjective. Since 2 is the only even prime number, the term odd prime refers to all prime numbers greater than 2.

The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers. Prime numbers have been the subject of intense research, yet some fundamental questions, such as the Riemann hypothesis and the Goldbach conjecture, have been unresolved for more than a century. The problem of modelling the distribution of prime numbers is a popular subject of investigation for number theorists: when looking at individual numbers, the primes seem to be randomly distributed, but the "global" distribution of primes follows well-defined laws.

The notion of prime number has been generalized in many different branches of mathematics.

• In ring theory, a branch of abstract algebra, the term "prime element" has a specific meaning. Here, a non-zero, non-unit ring element a is defined to be prime if whenever a divides b c for ring elements b and c, then a divides at least one of b or c. With this meaning, the additive inverse of any prime number is also prime. In other words, when considering the set of integers ( $\mathbb{Z}$ ) as a ring, − 7 is a prime element. Without further specification, however, "prime number" always means a positive integer prime. Among rings of complex algebraic integers, Eisenstein primes, and Gaussian primes may also be of interest.
• In knot theory, a prime knot is a knot which can not be disaggregated into a smaller prime knot.

In both the two above examples, the fundamental theorem of arithmetic (Every natural number can be 'uniquely' decomposed into a product of primes) does not apply.

History of prime numbers

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, efficient and still widely used method to compute primes.

After the Greeks, little happened until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibnitz and Euler). A special case of it may have been known long before by the Chinese. Other conjectures by Fermat turned out to be false, e.g. that all Fermat numbers are prime. The French monk Marin Mersenne made partially false assertions about Mersenne primes.

In 1747 Euler showed that all even perfect numbers are on Euclid's form. It is still unknown whether there exist odd perfect numbers. Euler's extensive work included many significant results about primes, e.g. that the infinite sum 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... is divergent.

Several mathematicians made contributions towards the prime number theorem until it was proved independently by Hadamard and de la Vallée Poussin in 1896.

Fermat's little theorem does not prove primality which can be determined by slow trial division for small numbers. Many mathematicians have worked on faster 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 test for Mersenne numbers (originated 1878), and the generalized Lucas–Lehmer test. More recent algorithms like APRT-CL, ECPP and AKS work on arbitrary numbers but remain much slower.

Until the 19th century most mathematicians considered the number 1 a prime, and there is still a large body of mathematical work that is valid despite labelling 1 a prime, such as the work of Stern and Zeisel. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. The change in label occurred so that it can be said 'each number has a unique factorization into primes.' See also "Arguments for and against the primality of 1".

For a long time, prime numbers were thought as having no possible application outside of number theory; 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 or the Diffie-Hellman key-exchange 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. 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.

Prime divisors

The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of one or more primes in a unique way, i.e. unique except for the order. The same prime may occur multiple times. Primes can thus be considered the "basic building blocks" of the natural numbers. For example, we can write

$23244 = 2^2 \times 3 \times 13 \times 149$

and any other factorization of 23244 as the product of primes will be identical except for the order of the factors. There are many prime factorization algorithms to do this in practice for larger numbers.

The importance of this theorem is one of the reasons for the exclusion of 1 from the set of prime numbers. If 1 were admitted as a prime, the precise statement of the theorem would require additional qualifications.

Properties of primes

• The base 10 numerals for all prime numbers except 2 and 5 end in 1, 3, 7, or 9. (Numerals ending in 0, 2, 4, 6 or 8 represent multiples of 2 and numerals ending in 5 represent multiples of 5, due to the use of the base 10 (2 × 5) system.)
• If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition was proved by Euclid and is known as Euclid's lemma. It is used in some proofs of the uniqueness of prime factorizations.
• The ring Z/nZ (see modular arithmetic) is a field if and only if n is a prime. Put another way: n is prime if and only if φ(n) = n − 1.
• If p is prime and a is any integer, then ap − a is divisible by p ( Fermat's little theorem).
• If p is a prime number other than 2 and 5, 1/p is always a recurring decimal, with a period of p-1 or a divisor of p-1. This can be deduced directly from Fermat's little theorem. 1/p expressed likewise in base q (i.e. other than base 10) has similar effect, provided that p is not a prime factor of q. The article on recurring decimals shows some of the interesting properties.
• An integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p ( Wilson's theorem). Conversely, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.
• If n is a positive integer greater than 1, then there is always a prime number p with n < p < 2n ( Bertrand's postulate).
• Adding the reciprocals of all primes together results in a divergent infinite series ( proof). 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 → ∞ (see Big O notation).
• In every arithmetic progression a, a + q, a + 2q, a + 3q,... where the positive integers a and q ≥ 1 are coprime, there are infinitely many primes ( Dirichlet's theorem on arithmetic progressions).
• The characteristic of every field is either zero or a prime number.
• 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. ( Sylow theorems)
• If p is prime and G is a group with pn elements, then G contains an element of order p.
• The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
• The Copeland-Erdős constant 0.235711131719232931374143..., obtained by concatenating the prime numbers, is known to be an irrational number.
• The value of the Riemann zeta function at each point in the complex plane is given as a meromorphic continuation of a function, defined by a product over the set of all primes for Re(s)>1:
$\zeta(s)= \sum_{n=1}^\infin \frac{1}{n^s} = \prod_{p} \frac{1}{1-p^{-s}}.$
Evaluating this identity at different integers provides an infinite number of products over the primes whose values can be calculated, the first two being
$\prod_{p} \frac{1}{1-p^{-1}} = \infty$
$\prod_{p} \frac{1}{1-p^{-2}}= \frac{\pi^2}{6}.$
• If $p>1\;$, the polynomial $x^{p-1}+x^{p-2}+ \cdots + 1$ is irreducible over Z / pZ if and only if $p\;$ is prime.
• If p is a prime number greater than 6 then p mod 6 is either 1 or 5 and p mod 30 is 1, 7, 11, 13, 17, 19, 23 or 29.

Classification of prime numbers

Two ways of classifying prime numbers, class n+ and class n-, were studied by Paul Erdős and John Selfridge.

Determining the class n+ of a prime number p involves looking at the largest prime factor of p + 1. If that largest prime factor is 2 or 3, then p is class 1+. But if that largest prime factor is another prime q, then the class n+ of p is one more than the class n+ of q. Sequences A005105 through A005108 list class 1+ through class 4+ primes.

The class n- is almost the same as class n+, except that the factorization of p - 1 is looked at instead.

The number of prime numbers

There are infinitely many prime numbers

The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Suppose you have a finite number of primes. Call this number m. Multiply all m primes together and add one (see Euclid number). The resulting number is not divisible by any of the finite set of primes, because dividing by any of these would give a remainder of one. And one is not divisible by any primes. Therefore it must either be prime itself, or be divisible by some other prime that was not included in the finite set. Either way, there must be at least m + 1 primes. But this argument applies no matter what m is; it applies to m + 1, too. So there are more primes than any given finite number.

This previous argument explains why the product of m primes plus 1 must be divisible by some prime not among the m primes, or be prime itself. A common mistake is thinking Euclid's proof says the prime product plus 1 is always prime.
(2 · 3 · 5 · 7 · 11 · 13) + 1 = 30,031 = 59 · 509 (both primes) shows this is not the case.

Other mathematicians have given their own proofs. One of those (due to Euler) shows that the sum of the reciprocals of all prime numbers diverges to infinity. Kummer's is particularly elegant and Harry Furstenberg provides one using general topology.

Counting the number of prime numbers below a given number

Even though the total number of primes is infinite, one could still ask "Approximately how many primes are there below 100,000?", or "How likely is a random 20-digit number to be prime?".

The prime counting function π(x) is defined as the number of primes up to x. There are known algorithms to compute exact values of π(x) faster than it would be to compute each prime up to x. Values as large as π(1020) can be calculated quickly and accurately with modern computers. Thus, e.g., π(100000) = 9592, and π(1020) = 2,220,819,602,560,918,840.

For larger values of x, beyond the reach of modern equipment, the prime number theorem provides a good estimate: π(x) is approximately x/ln(x). Even better estimates are known.

Location of prime numbers

Finding prime numbers

The Sieve of Eratosthenes is a simple way, and the Sieve of Atkin a fast way, to compute the list of all prime numbers up to a given limit.

In practice, though, one usually wants to check whether a given number is prime, rather than generate a list of primes. Further, it is often satisfactory to know the answer with a high probability. It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests. These typically pick a random number 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". Some of these tests are not perfect: there may be some composite numbers, called pseudoprimes for the respective test, that will be declared "probably prime" no matter what witness is chosen. However, the most popular probabilistic tests do not suffer from this drawback.

One method for determining whether a number is prime is to divide by all primes 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. One need not actually calculate the square root; once one sees that the quotient is less than the divisor, one can stop. This is known as trial division; it is the simplest primality test and it quickly becomes impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number-to-be-tested increases.

Pseudocode for programs to find primes

The least efficient algorithm starts out with a list containing just the number 2, then tries dividing each subsequent number by the primes already in the list. If it is not divisible evenly by any of them, it is added to the list. But if it is divisible evenly by any number already on the list, the program moves on to the next candidate.

The simplest, most elegant algorithm is perhaps the sieve of Eratosthenes.

// arbitrary search limit
limit ← 1000000

// assume all numbers are prime at first
is_prime(i) ← true, i ∈ [2, limit]

for n in [2, √limit]:
if is_prime(n):
// eliminate multiples of each prime,
// starting with its square
// 2n, 3n, ..., (n-1)n already eliminated
// nn, (n+1)n, (n+2)n, ... to be eliminated
is_prime(i) ← false, i ∈ {n², n²+n, n²+2n, ..., limit}

for n in [2, limit]:
if is_prime(n): print n



A more complicated, but more efficient algorithm (when properly optimized) is the sieve of Atkin. Its basic structure is as follows.

// arbitrary search limit
limit ← 1000000

// initialize the sieve
is_prime(i) ← false, i ∈ [5, limit]

// put in candidate primes:
// integers which have an odd number of
// representations by certain quadratic forms
for (x, y) in [1, √limit] × [1, √limit]:
n ← 4x²+y²
if (n ≤ limit) ∧ (n mod 12 = 1 ∨ n mod 12 = 5):
is_prime(n) ← ¬is_prime(n)
n ← 3x²+y²
if (n ≤ limit) ∧ (n mod 12 = 7):
is_prime(n) ← ¬is_prime(n)
n ← 3x²-y²
if (x > y) ∧ (n ≤ limit) ∧ (n mod 12 = 11):
is_prime(n) ← ¬is_prime(n)

// eliminate composites by sieving
for n in [5, √limit]:
if is_prime(n):
// n is prime, omit multiples of its square; this is sufficient because
// composites which managed to get on the list cannot be square-free
is_prime(k) ← false, k ∈ {n², 2n², 3n², ..., limit}

print 2, 3
for n in [5, limit]:
if is_prime(n): print n


Primality tests

A primality test algorithm is an algorithm which tests a number for primality, i.e. whether the number is a prime number.

• AKS primality test
• Fermat primality test
• Lucas-Lehmer test
• Solovay-Strassen primality test
• Miller-Rabin primality test

A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime. Probable primes which are in fact composite (such as Carmichael numbers) are called pseudoprimes.

In 2002, Indian scientists at IIT Kanpur discovered a new deterministic algorithm known as the AKS algorithm. The amount of time that this algorithm takes to check whether a number N is prime depends on a polynomial function of the number of digits of N (i.e. of the logarithm of N).

Formulas yielding prime numbers

There is no known formula for primes which is more efficient at finding primes than the methods mentioned above under "Finding prime numbers".

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. For example, the curious polynomial in one variable f(n) = n2 − n + 41 yields primes for n = 0,..., 40, but f(41) is composite. However, there are polynomials in several variables, whose positive values as the variables take all positive integer values are exactly the primes.

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

Special types of primes from formulas for 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 · 11 · ... of all the primes ≤ n. A prime is called factorial if it is of the form n! ± 1. The first factorial primes are:

n! − 1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166,... (sequence A002982 in OEIS)
n! + 1 is prime for n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154... (sequence A002981 in OEIS)

The largest known primorial prime is Π(392113) + 1, found by Heuer in 2001. The largest known factorial prime is 34790! − 1, found by Marchal, Carmody and Kuosa in 2002. It is not known whether there are infinitely many primorial or factorial primes.

Primes of the form 2n − 1 are known as Mersenne primes, while primes of the form $2^{2^n} + 1$ are known as Fermat primes. Prime numbers p where 2p + 1 is also prime are known as Sophie Germain primes. The following list is of other special types of prime numbers that come from formulas:

• Wieferich primes,
• Wilson primes,
• Wall-Sun-Sun primes,
• Wolstenholme primes,
• Unique primes,
• Newman-Shanks-Williams primes (NSW primes),
• Smarandache-Wellin primes,
• Wagstaff primes, and
• Supersingular primes.

Some primes are classified according to the properties of their digits in decimal or other bases. For example, numbers whose digits form a palindromic sequence are called palindromic primes, and a prime number is called a truncatable prime if successively removing the first digit at the left or the right yields only new prime numbers.

The distribution of the prime numbers

The distribution of all the prime numbers in the range of 1 to 76,800. Each pixel represents a number with black pixels meaning that number is prime and white ones not-prime.

The problem of modelling the distribution of prime numbers is a popular subject of investigation for number theorists. The prime numbers are distributed among the natural numbers in a (so far) unpredictable way, but there do appear to be laws governing their behaviour. 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. (Havil 2003, p. 163)

Paul Erdős said

God may not play dice with the universe, but something strange is going on with the prime numbers. [Referring to Albert Einstein's famous belief that "God does not play dice with the universe."]

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 behaviour, and that they obey these laws with almost military precision." (Havil 2003, p. 171)

Gaps between primes

Let pn denote the nth prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the difference between them, i.e.

gn = pn + 1pn.

We have g1 = 3 − 2 = 1, g2 = 5 − 3 = 2, g3 = 7 − 5 = 2, g4 = 11 − 7 = 4, and so on. The sequence (gn) of prime gaps has been extensively studied.

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. Therefore, there exist gaps between primes which are arbitrarily large, i.e. for any natural number N, there is an integer n with gn > N. (Choose n so that pn is the greatest prime number less than N! + 2.)

On the other hand, the gaps get arbitrarily small in proportion to the primes: the quotient gn/pn approaches zero as n approaches infinity. Note also that the twin prime conjecture asserts that gn = 2 for infinitely many integers n.

Location of the largest known prime

The largest known prime, as of September 2006, is 232,582,657 − 1 (this number is 9,808,358 digits long); it is the 44th known Mersenne prime. M32582657 was found on September 4, 2006 by Curtis Cooper and Steven Boone, professors at the University of Central Missouri (formerly Central Missouri State University) and members of a collaborative effort known as GIMPS. Before finding the prime, Cooper and Boone ran the GIMPS program on a peak of 700 CMSU computers for 9 years.

The next two largest known primes are also Mersenne Primes: M30,402,457 = 230,402,457 − 1 (43rd known Mersenne prime, 9,152,052 digits long) and M25964951 = 225,964,951 − 1 (42nd known Mersenne prime, 7,816,230 digits long). Historically, the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers, because there exists a particularly fast primality test for numbers of this form, the Lucas–Lehmer test for Mersenne numbers.

The largest known prime that is not a Mersenne prime is 27,653 × 29,167,433 + 1 (2,759,677 digits), a Proth number. This is also the seventh largest known prime of any form. It was found by the Seventeen or Bust project and it brings them one step closer to solving the Sierpinski problem.

Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].

Awards for finding primes

The Electronic Frontier Foundation (EFF) has offered a $100,000 (U.S.) prize to the first discoverers of a prime with at least 10 million digits. They also offer$150,000 for 100 million digits, and $250,000 for 1 billion digits. In 2000 they paid out$50,000 for 1 million digits.

The RSA Factoring Challenge offers prizes up to \$200,000 (U.S) for finding the prime factors of certain semiprimes of up to 2048 bits.

Generalizations of the prime concept

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics.

Prime elements in rings

One can define prime elements and irreducible elements in any integral domain. 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, ...}.

As an example, we consider the Gaussian integers Z[i], that is, complex numbers of the form a + bi with a and b in Z. This is an integral domain, and its prime elements are the Gaussian primes. Note that 2 is not a Gaussian prime, because it factors into the product of the two Gaussian primes (1 + i) and (1 − i). The element 3, however, remains prime in the Gaussian integers. In general, rational primes (i.e. prime elements in the ring Z of integers) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.

Prime ideals

In ring theory, one generally replaces the notion of number with that of ideal. Prime ideals 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), ...

A central problem in algebraic number theory is how a prime ideal factors when it is lifted to an extension field. For example, in the Gaussian integer example above, (2) ramifies into a prime power (1 + i and 1 − i generate the same prime ideal), prime ideals of the form (4k + 3) are inert (remain prime), and prime ideals of the form (4k + 1) split (are the product of 2 distinct prime ideals).

Primes in valuation theory

In class field theory yet another generalization is used. Given 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. With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.

Prime knots

In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. Specifically, it is one which cannot be written as the knot sum of two nontrivial knots.

Open questions

There are many open questions about prime numbers. A very significant one is the Riemann hypothesis, which 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.

Many famous conjectures appear to have a very high probability of being true (in a formal sense, many of them follow from simple heuristic probabilistic arguments):

• Prime Euclid numbers: It is not known whether or not there are an infinite number of prime Euclid numbers.
• Strong Goldbach conjecture: Every even integer greater than 2 can be written as a sum of two primes.
• Weak Goldbach conjecture: Every odd integer greater than 5 can be written as a sum of three primes.
• Twin prime conjecture: There are infinitely many twin primes, pairs of primes with difference 2.
• Polignac's conjecture: For every positive integer n, there are infinitely many pairs of consecutive primes which differ by 2n. When n = 1 this is the twin prime conjecture.
• A weaker form of Polignac's conjecture: Every even number is the difference of two primes.
• It is widely believed there are infinitely many Mersenne primes, but not Fermat primes.
• Many believe there are infinitely many primes of the form n2 + 1.
• Many well-known conjectures are special cases of the broad Schinzel's hypothesis H.
• Many believe there are infinitely many Fibonacci primes.
• Legendre's conjecture: There is a prime number between n2 and (n + 1)2 for every positive integer n.
• Cramér's conjecture: $\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1$. This conjecture implies Legendre's, but its status is more unsure.
• Brocard's conjecture: There are always at least four primes between the squares of successive primes > 2.

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. 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.

Public-key cryptography

Several public-key cryptography algorithms, such as RSA, are based on large prime numbers (that is, greater than 10100).

Prime numbers in nature

Many numbers occur in nature, and inevitably some of these are prime. There are, however, relatively few examples of numbers that appear in nature because they are prime. For example, most starfish have 5 arms, and 5 is a prime number. However there is no evidence to suggest that starfish have 5 arms because 5 is a prime number. Indeed, some starfish have different numbers of arms. Echinaster luzonicus normally has six arms, Luidia senegalensis has nine arms, and Solaster endeca can have as many as twenty arms. Why the majority of starfish (and most other echinoderms) have five-fold symmetry remains a mystery.

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

Primes in pop culture

• In Carl Sagan's novel Contact a message from outer space is sent to humanity. This message is hashed into pieces and comes in intervals of incremented prime numbers from 2 to 101. The same device was later used in an episode of Star Trek: The Next Generation.

(These fictional alien transmissions might have been inspired by the use of prime numbers in constructing the Arecibo message transmitted in 1974).

• In Stephen King's novels The Waste Lands and Wizard and Glass, the protagonists solve a puzzle using primes in order to ride on the sentient and deranged train, Blaine the Mono.
• In the Stargate Atlantis episode "Hot Zone", physicists Rodney McKay and Radek Zelenka play the game "Prime or Not Prime" in which players randomly name numbers and expect the other player or players to determine whether the number is prime.
• In the film Cube, the protagonists discover that prime numbers encoded on passageways represent a possible key to their escape from a mysterious facility filled with lethal booby traps.

Trivia

As a publicity stunt against the Digital Millennium Copyright Act and other WIPO Copyright Treaty implementations, some people have tried to encode various forms of the DeCSS program in such a way that the executable program code becomes a prime number, thus creating illegal primes. Such numbers, when converted to binary and executed as a computer program, perform acts encumbered by applicable law in one or more jurisdictions.

The largest known prime consisting of a prime number of decimal digits is 27,653 · 29,167,433 + 1 which has 2,759,677 digits. As of January 2007 the 6 larger known primes (all Mersenne primes) have numbers of digits which are even or divisible by 3.

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