Prime number
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, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) 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 their order.
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
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 large 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 that differ by two. Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which 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 elements and prime ideals.
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 numbers.[2] In other words, is prime if items cannot be divided up into smaller equal-size groups of more than one item,[3] 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.[4] For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,[5] 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. 4 = 2 × 2 and 6 = 2 × 3 are both composite.
The divisors of a natural number are the natural numbers that divide evenly. Every natural number has both 1 and itself as a divisor. If it has any other divisor, it cannot be prime. This leads to an equivalent definition of prime numbers: they are the numbers with exactly two positive divisors. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition.[6] 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 divides evenly.[7]
The first 25 prime numbers (all the prime numbers less than 100) are:[8]
- 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 (sequence A000040 in the OEIS).
No
The set of all primes is sometimes denoted by (a
History
The
Around 1000 AD, the
In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler).[18] Fermat also investigated the primality of the Fermat numbers ,[19] and Marin Mersenne studied the Mersenne primes, prime numbers of the form with itself a prime.[20] Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.[21] Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes.[14] 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 .[22] At the start of the 19th century, Legendre and Gauss conjectured that as tends to infinity, the number of primes up to is asymptotic to , where is the natural logarithm of . A weaker consequence of this high density of primes was Bertrand's postulate, that for every there is a prime between and , proved in 1852 by
Many mathematicians have worked on primality tests 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),[26] Proth's theorem (c. 1878),[27] the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.[16]
Since 1951 all the
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.[15][33][34] 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.[35]
Primality of one
Most early Greeks did not even consider 1 to be a number,
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 any number of copies of 1.
Elementary properties
Unique factorization
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.[44] 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.[45] 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.[46]
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).[47] 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.[48]
Infinitude
There are
- 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's proof[52] 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.[53] 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
Other examples of prime-generating formulas come from
are prime for any natural number in the first formula, and any number of exponents in the second formula.[56] Here represents the
Open questions
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.[57] 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.[58] As of 2014[update], this conjecture has been verified for all numbers up to [59] 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.[60] 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).[61] Also, any even integer greater than 10 can be written as the sum of six primes.[62] The branch of number theory studying such questions is called additive number theory.[63]
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 [64] However, large prime gaps occur much earlier than this argument shows.[65] For example, the first prime gap of length 8 is between the primes 89 and 97,[66] much smaller than It is conjectured that there are infinitely many
Analytic properties
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 .[71] 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).[72]
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 .[73] 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.[74] 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.[75] 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
Number of primes below a given bound
The prime-counting function is defined as the number of primes not greater than .[76] 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 .[77] The prime number theorem states that is asymptotic to , which is denoted as
and means that the ratio of to the right-hand fraction
Arithmetic progressions
An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.[81] This difference is called the modulus of the progression.[82] 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.[83]
The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.[35][84]
Prime values of quadratic polynomials
Euler noted that the function
yields prime numbers for , although composite numbers appear among its later 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.[88]
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.[89] 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.[90] 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.[91]
The Riemann hypothesis states that the
Abstract algebra
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 .[97] 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.[98] Equality of integers corresponds to congruence in modular arithmetic: and are congruent (written mod ) when they have the same remainder after division by .[99] 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.[100]
Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if (mod ), then (mod ).[101] Summing this over all choices of gives the equation
valid whenever is prime.
p-adic numbers
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.[104][105]
This picture of an order, absolute value, and complete field derived from them can be generalized to
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.[108]
The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers , the ring of complex numbers of the form where denotes the imaginary unit and and are arbitrary integers. Its prime elements are known as
Prime ideals
Not every ring is a unique factorization domain. For instance, in the ring of numbers (for integers and ) the number has two factorizations , where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an ideal, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements. Prime ideals, which generalize prime elements in the sense that the
The
Group theory
In the theory of finite groups the Sylow theorems imply that, if a power of a prime number divides the
Computational methods
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 mathematics[b] other than the use of prime numbered gear teeth to distribute wear evenly.[117] In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.[118]
This vision of the purity of number theory 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.[32] These applications have led to significant study of algorithms for computing with prime numbers, and in particular of primality testing, methods for determining whether a given number is prime. The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types. Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments (or all of its prime factors) are called factorization algorithms. Prime numbers are also used in computing for checksums, hash tables, and pseudorandom number generators.
Trial division
The most basic method of checking the primality of a given integer is called trial division. This method divides by each integer from 2 up to the square root of . Any such integer dividing evenly establishes as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever , one of the two factors and is less than or equal to the square root of . Another optimization is to check only primes as factors in this range.[119] For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to , which are 2, 3, and 5. Each division produces a nonzero remainder, so 37 is indeed prime.
Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers.[120] However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter.[121]
Sieves
Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed.[122] The oldest method for generating a list of primes is called the sieve of Eratosthenes.[123] The animation shows an optimized variant of this method.[124] Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin.[125] In advanced mathematics, sieve theory applies similar methods to other problems.[126]
Primality testing versus primality proving
Some of the fastest modern tests for whether an arbitrary given number is prime are
In contrast, some other algorithms guarantee that their answer will always be correct: primes will always be determined to be prime and composites will always be determined to be composite. For instance, this is true of trial division. The algorithms with guaranteed-correct output include both deterministic (non-random) algorithms, such as the AKS primality test,[130] and randomized
The following table lists some of these tests. Their running time is given in terms of , the number to be tested and, for probabilistic algorithms, the number of tests performed. Moreover, is an arbitrarily small positive number, and log is the
Test | Developed in | Type | Running time | Notes | References |
---|---|---|---|---|---|
AKS primality test | 2002 | deterministic | [130][133] | ||
Elliptic curve primality proving
|
1986 | Las Vegas | heuristically | [132] | |
Baillie–PSW primality test | 1980 | Monte Carlo | [134][135] | ||
Miller–Rabin primality test | 1980 | Monte Carlo | error probability | [136] | |
Solovay–Strassen primality test | 1977 | Monte Carlo | error probability | [136] |
Special-purpose algorithms and the largest known prime
In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the
The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits.[140] The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.[141]
Type | Prime | Number of decimal digits | Date | Found by |
---|---|---|---|---|
Mersenne prime | 282,589,933 − 1 | 24,862,048 | December 7, 2018[1] | Patrick Laroche, Great Internet Mersenne Prime Search |
Proth prime | 10,223 × 231,172,165 + 1 | 9,383,761 | October 31, 2016[142] | Péter Szabolcs, PrimeGrid[143] |
factorial prime | 208,003! − 1 | 1,015,843 | July 2016 | Sou Fukui[144] |
primorial prime[e] | 1,098,133# − 1 | 476,311 | March 2012 | James P. Burt, PrimeGrid[146] |
twin primes | 2,996,863,034,895 × 21,290,000 ± 1 | 388,342 | September 2016 | Tom Greer, PrimeGrid[147] |
Integer factorization
Given a composite integer , the task of providing one (or all) prime factors is referred to as factorization of . It is significantly more difficult than primality testing,[148] and although many factorization algorithms are known, they are slower than the fastest primality testing methods. Trial division and Pollard's rho algorithm can be used to find very small factors of ,
Other computational applications
Several
Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers. Carter and Wegman generalized this method to -independent hashing by using higher-degree polynomials, again modulo large primes.[156] As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table.[157]
Some
Other applications
Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line, or so that every triangle formed by three of the points has large area.[162] Another example is Eisenstein's criterion, a test for whether a polynomial is irreducible based on divisibility of its coefficients by a prime number and its square.[163]
The concept of a 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
Beyond mathematics and computing, prime numbers have potential connections to
Constructible polygons and polygon partitions
with a
It is possible to partition any convex polygon into smaller convex polygons of equal area and equal perimeter, when is a power of a prime number, but this is not known for other values of .[171]
Quantum mechanics
Beginning with the work of
Biology
The evolutionary strategy used by
Arts and literature
Prime numbers have influenced many artists and writers. The French
In his science fiction novel Contact, scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975.[181] In the novel The Curious Incident of the Dog in the Night-Time by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers as a way to convey the mental state of its main character, a mathematically gifted teen with Asperger syndrome.[182] 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.[183]
Notes
- ^ A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.[28]
- ^ a b For instance, Beiler writes that number theorist Ernst Kummer loved his ideal numbers, closely related to the primes, "because they had not soiled themselves with any practical applications",[30] and Katz writes that Edmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as geometry that had already shown themselves to be useful.[31]
- ^ In this test, the term is negative if is a square modulo the given (supposed) prime , and positive otherwise. More generally, for non-prime values of , the term is the (negated) Jacobi symbol, which can be calculated using quadratic reciprocity.
- ^ Indeed, much of the analysis of elliptic curve primality proving is based on the assumption that the input to the algorithm has already passed a probabilistic test.[131]
- ^ The primorial function of , denoted by , yields the product of the prime numbers up to , and a primorial prime is a prime of one of the forms .[145]
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No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years.
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External links
- "Prime number". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
- Caldwell, Chris, The Prime Pages at primes.utm.edu.
- Prime Numbers on In Our Time at the BBC
- 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.
Generators and calculators
- Prime factors calculator can factorize any positive integer up to 20 digits.
- Fast Online primality test with factorization makes use of the Elliptic Curve Method (up to thousand-digits numbers, requires Java).
- Huge database of prime numbers
- Prime Numbers up to 1 trillion Archived 2021-02-27 at the Wayback Machine