A prime number (or a prime) is a natural number
greater than 1 that is not a product
of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number
. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself.
However, 4 is composite because it is a product () in which both numbers are smaller than 4. Primes are central in number theory
because of the fundamental theorem of arithmetic
: every natural number greater than 1 is either a prime itself or can be factorized
as a product of primes that is unique up to
The property of being prime is called primality. A simple but slow method of checking the primality of a given number
, called trial division
, tests whether
is a multiple of any integer between 2 and
. Faster algorithms include the Miller–Rabin primality test
, which is fast but has a small chance of error, and the AKS primality test
, which always produces the correct answer in polynomial time
but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne number
s. the largest known prime number
is a Mersenne prime with 24,862,048 decimal digits
There are infinitely many
primes, as demonstrated by Euclid
around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem
, proven at the end of the 19th century, which says that the probability
of a randomly chosen number being prime is inversely proportional
to its number of digits, that is, to its logarithm
Several historical questions regarding prime numbers are still unsolved. These include Goldbach's conjecture
, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime
conjecture, that there are infinitely many pairs of primes having just one even number between them. Such questions spurred the development of various branches of number theory, focusing on analytic
aspects of numbers. Primes are used in several routines in information technology
, such as public-key cryptography
, which relies on the difficulty of factoring
large numbers into their prime factors. In abstract algebra
, objects that behave in a generalized way like prime numbers include prime element
s and prime ideal
Definition and examples
A natural number
(1, 2, 3, 4, 5, 6, etc.) is called a ''prime number'' (or a ''prime'') if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called composite number
s. In other words,
is prime if
items cannot be divided up into smaller equal-size groups of more than one item, or if it is not possible to arrange
dots into a rectangular grid that is more than one dot wide and more than one dot high.
For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers, as there are no other numbers that divide them evenly (without a remainder).
1 is not prime, as it is specifically excluded in the definition. and are both composite.
s of a natural number
are the natural numbers that divide
Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This idea leads to a different but equivalent definition of the primes: they are the numbers with exactly two positive divisor
s, 1 and the number itself.
Yet another way to express the same thing is that a number
is prime if it is greater than one and if none of the numbers
The first 25 prime numbers (all the prime numbers less than 100) are:
No even number
greater than 2 is prime because any such number can be expressed as the product
. Therefore, every prime number other than 2 is an odd number
, and is called an ''odd prime''. Similarly, when written in the usual decimal
system, all prime numbers larger than 5 end in 1, 3, 7, or 9. The numbers that end with other digits are all composite:
decimal numbers that end in 0, 2, 4, 6, or 8 are even, and decimal numbers that end in 0 or 5 are divisible by 5.
of all primes is sometimes denoted by
capital ''P'') or by
(a blackboard bold
The Rhind Mathematical Papyrus
, from around 1550 BC, has Egyptian fraction
expansions of different forms for prime and composite numbers. However, the earliest surviving records of the explicit study of prime numbers come from ancient Greek mathematics
'' (c. 300 BC) proves the infinitude of primes
and the fundamental theorem of arithmetic
, and shows how to construct a perfect number
from a Mersenne prime
Another Greek invention, the Sieve of Eratosthenes
, is still used to construct lists of primes.
Around 1000 AD, the Islamic
mathematician Ibn al-Haytham
(Alhazen) found Wilson's theorem
, characterizing the prime numbers as the numbers
that evenly divide
. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it. Another Islamic mathematician, Ibn al-Banna' al-Marrakushi
, observed that the sieve of Eratosthenes can be sped up by testing only the divisors up to the square root of the largest number to be tested. Fibonacci
brought the innovations from Islamic mathematics back to Europe. His book ''Liber Abaci
'' (1202) was the first to describe trial division
for testing primality, again using divisors only up to the square root.
In 1640 Pierre de Fermat
stated (without proof) Fermat's little theorem
(later proved by Leibniz
). Fermat also investigated the primality of the Fermat number
, and Marin Mersenne
studied the Mersenne prime
s, prime numbers of the form
itself a prime. Christian Goldbach
formulated Goldbach's conjecture
, that every even number is the sum of two primes, in a 1742 letter to Euler. Euler proved Alhazen's conjecture (now the Euclid–Euler theorem
) that all even perfect numbers can be constructed from Mersenne primes.
He introduced methods from mathematical analysis
to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes
At the start of the 19th century, Legendre and Gauss conjectured that as
tends to infinity, the number of primes up to
is the natural logarithm
. A weaker consequence of this high density of primes was Bertrand's postulate
, that for every
there is a prime between
, proved in 1852 by Pafnuty Chebyshev
. Ideas of Bernhard Riemann
in his 1859 paper on the zeta-function
sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis
remains unproven, Riemann's outline was completed in 1896 by Hadamard
and de la Vallée Poussin
, and the result is now known as the prime number theorem
. Another important 19th century result was Dirichlet's theorem on arithmetic progressions
, that certain arithmetic progression
s contain infinitely many primes.
Many mathematicians have worked on primality test
s for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test
for Fermat numbers (1877), Proth's theorem
(c. 1878), the Lucas–Lehmer primality test
(originated 1856), and the generalized Lucas primality test
Since 1951 all the largest known prime
s have been found using these tests on computer
s. The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search
and other distributed computing
[ The idea that prime numbers had few applications outside of pure mathematics was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.]
The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form. The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size. [, pp. 18, 47.]
Primality of one
Most early Greeks did not even consider 1 to be a number,
[ For a selection of quotes from and about the ancient Greek positions on this issue, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6.] so they could not consider its primality. A few mathematicians from this time also considered the prime numbers to be a subdivision of the odd numbers, so they also did not consider 2 to be prime. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.
By the Middle Ages and Renaissance mathematicians began treating 1 as a number, and some of them included it as the first prime number. In the mid-18th century Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler; however, Euler himself did not consider 1 to be prime. In the 19th century many mathematicians still considered 1 to be prime, and lists of primes that included 1 continued to be published as recently as 1956.
If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1. By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".
Writing a number as a product of prime numbers is called a ''prime factorization'' of the number. For example:
The terms in the product are called ''prime factors''. The same prime factor may occur more than once; this example has two copies of the prime factor When a prime occurs multiple times, exponentiation can be used to group together multiple copies of the same prime number: for example, in the second way of writing the product above, denotes the square or second power of
The central importance of prime numbers to number theory and mathematics in general stems from the ''fundamental theorem of arithmetic''. This theorem states that every integer larger than 1 can be written as a product of one or more primes. More strongly,
this product is unique in the sense that any two prime factorizations of the same number will have the same numbers of copies of the same primes,
although their ordering may differ. So, although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes can thus be considered the "basic building blocks" of the natural numbers.
Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If is a prime number and divides a product of integers and then divides or divides (or both). Conversely, if a number has the property that when it divides a product it always divides at least one factor of the product, then must be prime.
There are infinitely many prime numbers. Another way of saying this is that the sequence
:2, 3, 5, 7, 11, 13, ...
of prime numbers never ends. This statement is referred to as ''Euclid's theorem'' in honor of the ancient Greek mathematician Euclid, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on Fermat numbers, Furstenberg's proof using general topology, and Kummer's elegant proof.
Euclid's proof shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add If the list consists of the primes this gives the number
By the fundamental theorem, has a prime factorization
with one or more prime factors. is evenly divisible by each of these factors, but has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of can be in the given list. Because there is no finite list of all the primes, there must be infinitely many primes.
The numbers formed by adding one to the products of the smallest primes are called Euclid numbers. The first five of them are prime, but the sixth,
is a composite number.
Formulas for primes
There is no known efficient formula for primes. For example, there is no non-constant polynomial, even in several variables, that takes ''only'' prime values.
However, there are numerous expressions that do encode all primes, or only primes. One possible formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once. There is also a set of Diophantine equations in nine 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.
Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants and such that
are prime for any natural number in the first formula, and any number of exponents in the second formula. Here represents the floor function, the largest integer less than or equal to the number in question. However, these are not useful for generating primes, as the primes must be generated first in order to compute the values of or
Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer greater than 2 can be written as a sum of two primes. , this conjecture has been verified for all numbers up to Weaker statements than this have been proven, for example, Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). Also, any even integer greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.
Another type of problem concerns prime gaps, the differences between consecutive primes.
The existence of arbitrarily large prime gaps can be seen by noting that the sequence consists of composite numbers, for any natural number However, large prime gaps occur much earlier than this argument shows.
For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2; this is the twin prime conjecture. Polignac's conjecture states more generally that for every positive integer there are infinitely many pairs of consecutive primes that differ by [, Gaps between primes, pp. 186–192.]
Andrica's conjecture, Brocard's conjecture, [, p. 183.] Legendre's conjecture, [ Note that Chan lists Legendre's conjecture as "Sierpinski's Postulate".] and Oppermann's conjecture all suggest that the largest gaps between primes from to should be at most approximately a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at Prime gaps can be generalized to prime -tuples, patterns in the differences between more than two prime numbers. Their infinitude and density are the subject of the first Hardy–Littlewood conjecture, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.
Analytic number theory studies number theory through the lens of continuous functions, limits, infinite series, and the related mathematics of the infinite and infinitesimal.
This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem.
The problem asked for the value of the infinite sum
which today can be recognized as the value of the Riemann zeta function. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis. Euler showed that .
The reciprocal of this number, , is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common).
The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the -th prime is known.
Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
with relatively prime integers and take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same have approximately the same proportions of primes.
Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often.
Analytical proof of Euclid's theorem
Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes,
Euler showed that, for any arbitrary real number , there exists a prime for which this sum is bigger than . This shows that there are infinitely many primes, because if there were finitely many primes the sum would reach its maximum value at the biggest prime rather than growing past every .
The growth rate of this sum is described more precisely by Mertens' second theorem. For comparison, the sum
does not grow to infinity as goes to infinity (see the Basel problem). In this sense, prime numbers occur more often than squares of natural numbers,
although both sets are infinite.
Brun's theorem states that the sum of the reciprocals of twin primes,
is finite. Because of Brun's theorem, it is not possible to use Euler's method to solve the twin prime conjecture, that there exist infinitely many twin primes.
Number of primes below a given bound
The prime counting function is defined as the number of primes not greater than . For example, , since there are five primes less than or equal to 11. Methods such as the Meissel–Lehmer algorithm can compute exact values of faster than it would be possible to list each prime up to . The prime number theorem states that is asymptotic to , which is denoted as
and means that the ratio of to the right-hand fraction approaches 1 as grows to infinity.
This implies that the likelihood that a randomly chosen number less than is prime is (approximately) inversely proportional to the number of digits in .
It also implies that the th prime number is proportional to
and therefore that the average size of a prime gap is proportional to .
A more accurate estimate for is given by the offset logarithmic integral
Large gaps between consecutive primes
, pp. 78–79.
An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference. This difference is called the modulus of the progression. For example,
:3, 12, 21, 30, 39, ...,
is an infinite arithmetic progression with modulus 9. In an arithmetic progression, all the numbers have the same remainder when divided by the modulus; in this example, the remainder is 3. Because both the modulus 9 and the remainder 3 are multiples of 3, so is every element in the sequence. Therefore, this progression contains only one prime number, 3 itself. In general, the infinite progression
can have more than one prime only when its remainder and modulus are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.
The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.
Prime values of quadratic polynomials
Euler noted that the function
yields prime numbers for , although composite numbers appear among its later values. The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients
in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.
The Ulam spiral arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.
Zeta function and the Riemann hypothesis
One of the most famous unsolved questions in mathematics, dating from 1859, and one of the Millennium Prize Problems, is the Riemann hypothesis, which asks where the zeros of the Riemann zeta function are located.
This function is an analytic function on the complex numbers. For complex numbers with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers,
This equality between a sum and a product, discovered by Euler, is called an Euler product. The Euler product can be derived from the fundamental theorem of arithmetic, and shows the close connection between the zeta function and the prime numbers.
It leads to another proof that there are infinitely many primes: if there were only finitely many,
then the sum-product equality would also be valid at , but the sum would diverge (it is the harmonic series ) while the product would be finite, a contradiction.
The Riemann hypothesis states that the zeros of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2. The original proof of the prime number theorem was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1,
/ref> although other more elementary proofs have been found.
The prime-counting function can be expressed by Riemann's explicit formula as a sum in which each term comes from one of the zeros of the zeta function; the main term of this sum is the logarithmic integral, and the remaining terms cause the sum to fluctuate above and below the main term.
In this sense, the zeros control how regularly the prime numbers are distributed. If the Riemann hypothesis is true, these fluctuations will be small, and the
asymptotic distribution of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of for intervals near a number ).
Modular arithmetic and finite fields
Modular arithmetic modifies usual arithmetic by only using the numbers , for a natural number called the modulus.
Any other natural number can be mapped into this system by replacing it by its remainder after division by .
Modular sums, differences and products are calculated by performing the same replacement by the remainder
on the result of the usual sum, difference, or product of integers. Equality of integers corresponds to ''congruence'' in modular arithmetic:
and are congruent (written mod ) when they have the same remainder after division by . However, in this system of numbers, division by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number as modulus, division by is possible: , because clearing denominators by multiplying both sides by gives the valid formula . However, with the composite modulus , division by is impossible. There is no valid solution to : clearing denominators by multiplying by causes the left-hand side to become while the right-hand side becomes either or .
In the terminology of abstract algebra, the ability to perform division means that modular arithmetic modulo a prime number forms a field or, more specifically, a finite field, while other moduli only give a ring but not a field.
Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if
(mod ), then (mod ).
Summing this over all choices of gives the equation
valid whenever is prime.
Giuga's conjecture says that this equation is also a sufficient condition for to be prime.
Wilson's theorem says that an integer is prime if and only if the factorial is congruent to mod . For a composite this cannot hold, since one of its factors divides both and , and so is impossible.
The -adic order of an integer is the number of copies of in the prime factorization of . The same concept can be extended from integers to rational numbers by defining the -adic order of a fraction to be . The -adic absolute value of any rational number is then defined as
. Multiplying an integer by its -adic absolute value cancels out the factors of in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their -adic distance, the -adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of . In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a complete field, the rational numbers with the -adic distance can be extended to a different complete field, the -adic numbers.
This picture of an order, absolute value, and complete field derived from them can be generalized to algebraic number fields and their valuations (certain mappings from the multiplicative group of the field to a totally ordered additive group, also called orders), absolute values (certain multiplicative mappings from the field to the real numbers, also called norms), [ See also p. 64.] and places (extensions to complete fields in which the given field is a dense set, also called completions). The extension from the rational numbers to the real numbers, for instance, is a place in which the distance between numbers is the usual absolute value of their difference. The corresponding mapping to an additive group would be the logarithm of the absolute value, although this does not meet all the requirements of a valuation. According to Ostrowski's theorem, up to a natural notion of equivalence, the real numbers and -adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers. The local-global principle allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.
Prime elements in rings
A commutative ring is an algebraic structure where addition, subtraction and multiplication are defined. The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, ''prime elements'' and ''irreducible elements''. An element of a ring is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a unit), and satisfies the following requirement: whenever divides the product of two elements of , it also divides at least one of or . An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set,
In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for unique factorization domains.
The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers