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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, Template:Nowrap or Template:Nowrap, 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 Template:Tmath, called trial division, tests whether Template:Tmath is a multiple of any integer between 2 and Template:Tmath. 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 numbers. Template:As of the largest known prime number is a Mersenne prime with 41,024,320 decimal digits.<ref name="GIMPS-2024">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal</ref>
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 roughly 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 examplesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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.<ref>Template:Cite book</ref> In other words, Template:Tmath is prime if Template:Tmath items cannot be divided up into smaller equal-size groups of more than one item,<ref>Template:Cite book</ref> or if it is not possible to arrange Template:Tmath dots into a rectangular grid that is more than one dot wide and more than one dot high.<ref>Template:Cite book</ref> For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,<ref>Template:Cite book</ref> 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. Template:Nowrap and Template:Nowrap are both composite.
The divisors of a natural number Template:Tmath are the natural numbers that divide Template:Tmath 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.<ref>Template:Cite book</ref> Yet another way to express the same thing is that a number Template:Tmath is prime if it is greater than one and if none of the numbers <math>2, 3, \dots, n-1</math> divides Template:Tmath evenly.<ref>Template:Cite book</ref>
The first 25 prime numbers (all the prime numbers less than 100) are:<ref name=ziegler>Template:Cite journal</ref>
- 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 even number Template:Tmath greater than 2 is prime because any such number can be expressed as the product Template:Tmath. Therefore, every prime number other than 2 is an odd number, and is called an odd prime.<ref>Template:Cite book</ref> 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.<ref>Template:Cite book</ref>
The set of all primes is sometimes denoted by <math>\mathbf{P}</math> (a boldface capital P)<ref>Template:Cite book</ref> or by <math>\mathbb{P}</math> (a blackboard bold capital P).<ref>Template:Cite book</ref>
HistoryEdit
The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers.<ref>Bruins, Evert Marie, review in Mathematical Reviews of Template:Cite journal</ref> However, the earliest surviving records of the study of prime numbers come from the ancient Greek mathematicians, who called them Template:Transliteration ({{#invoke:Lang|lang}}). Euclid's Elements (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.<ref name="stillwell-2010-p40">Template:Cite book</ref> Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of Template:Nowrap
Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem, characterizing the prime numbers as the numbers Template:Tmath that evenly divide Template:Tmath. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.<ref>Template:MacTutor Biography</ref> Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit.<ref name="mollin"/> Fibonacci took the innovations from Islamic mathematics 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.<ref name="mollin"/>
In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler).<ref>Template:Harvnb, 8. Fermat's Little Theorem (November 2003), p. 45</ref> Fermat also investigated the primality of the Fermat numbers Template:Tmath,<ref>Template:Cite book</ref> and Marin Mersenne studied the Mersenne primes, prime numbers of the form <math>2^p-1</math> with Template:Tmath itself a prime.<ref>Template:Cite book</ref> Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.<ref>Template:Cite book</ref> Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes.<ref name="stillwell-2010-p40"/> 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 Template:Tmath.<ref>Template:Cite book</ref> At the start of the 19th century, Legendre and Gauss conjectured that as Template:Tmath tends to infinity, the number of primes up to Template:Tmath is asymptotic to Template:Tmath, where <math>\log x</math> is the natural logarithm of Template:Tmath. A weaker consequence of this high density of primes was Bertrand's postulate, that for every <math>n > 1</math> there is a prime between Template:Tmath and Template:Tmath, proved in 1852 by Pafnuty Chebyshev.<ref>Template:Cite journal. (Proof of the postulate: 371–382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp. 15–33, 1854</ref> 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.<ref>Template:Cite book</ref> Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.<ref>Template:Cite book</ref>
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),<ref>Template:Cite book</ref> Proth's theorem (c. 1878),<ref>Template:Cite book</ref> the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.<ref name="mollin"/>
Since 1951 all the largest known primes have been found using these tests on computers.Template:Efn The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects.<ref name=ziegler/><ref>Template:Harvnb, p. 245.</ref> The idea that prime numbers had few applications outside of pure mathematicsTemplate:Efn was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.<ref name="ent-7">Template:Cite book</ref>
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.<ref name="pomerance-sciam"/><ref>Template:Cite book</ref><ref>Template:Cite book</ref> 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.<ref name="neale-18-47">Template:Harvnb, pp. 18, 47.</ref>
Primality of oneEdit
Most early Greeks did not even consider 1 to be a number,<ref name="crxk-34">Template:Cite journal For a selection of quotes from and about the ancient Greek positions on the status of 1 and 2, see in particular pp. 3–4. For the Islamic mathematicians, see p. 6.</ref><ref>Template:Cite book</ref> so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including Nicomachus, Iamblichus, Boethius, and Cassiodorus, also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider Template:Tmath to be prime either. However, Euclid and a majority of the other Greek mathematicians considered Template:Tmath as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.<ref name="crxk-34"/> By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and by the 17th century some of them included it as the first prime number.<ref>Template:Harvnb, pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet.</ref> In the mid-18th century, Christian Goldbach listed 1 as prime in his correspondence with Leonhard Euler;<ref>Template:Harvnb, pp. 6–7.</ref> however, Euler himself did not consider 1 to be prime.<ref>Template:Harvnb, p. 15.</ref> Many 19th century mathematicians still considered 1 to be prime,<ref name="cx"/> and Derrick Norman Lehmer included 1 in his list of primes less than ten million published in 1914.Template:Sfn Lists of primes that included 1 continued to be published as recently Template:Nowrap However, 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".<ref name="cx"/>
If 1 were to be considered a prime, many statements involving primes would need to be awkwardly reworded. 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.<ref name="cx">Template:Cite journal</ref> 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.<ref name="cg-bon-129-130"/> 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.<ref>For the totient, see Template:Harvnb, p. 245. For the sum of divisors, see Template:Cite book</ref>
Elementary propertiesEdit
Unique factorizationEdit
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Writing a number as a product of prime numbers is called a prime factorization of the number. For example:
- <math>\begin{align}
50 &= 2\times 5\times 5\\
&=2\times 5^2.
\end{align}</math>
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 <math>5.</math> 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, <math>5^2</math> denotes the square or second power of Template:Tmath.
The central importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic.<ref>Template:Cite book</ref> 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.<ref>Template:Harvnb, Section 2, Theorem 2, p. 16; Template:Cite book</ref> 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.<ref>Template:Cite book</ref>
Some proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If Template:Tmath is a prime number and Template:Tmath divides a product <math>ab</math> of integers Template:Tmath and <math>b,</math> then Template:Tmath divides Template:Tmath or Template:Tmath divides Template:Tmath (or both).<ref>Template:Harvnb, Section 2, Lemma 5, p. 15; Template:Cite book</ref> Conversely, if a number Template:Tmath has the property that when it divides a product it always divides at least one factor of the product, then Template:Tmath must be prime.<ref>Template:Cite book</ref>
InfinitudeEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} There are infinitely many prime numbers. Another way of saying this is that the sequence
- <math>2, 3, 5, 7, 11, 13, ...</math>
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,<ref>Letter in Latin from Goldbach to Euler, July 1730.</ref> Furstenberg's proof using general topology,<ref>Template:Cite journal </ref> and Kummer's elegant proof.<ref>Template:Cite book</ref>
Euclid's proof<ref>Euclid's Elements, Book IX, Proposition 20. See David Joyce's English translation of Euclid's proof or Template:Cite book</ref> shows that every finite list of primes is incomplete. The key idea is to multiply together the primes in any given list and add <math>1.</math> If the list consists of the primes <math>p_1,p_2,\ldots, p_n,</math> this gives the number
- <math> N = 1 + p_1\cdot p_2\cdots p_n. </math>
By the fundamental theorem, Template:Tmath has a prime factorization
- <math> N = p'_1\cdot p'_2\cdots p'_m </math>
with one or more prime factors. Template:Tmath is evenly divisible by each of these factors, but Template:Tmath has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of Template:Tmath 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.<ref>Template:Cite book</ref> The first five of them are prime, but the sixth,
- <math>1+\big(2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\big) = 30031 = 59\cdot 509,</math>
is a composite number.
Formulas for primesEdit
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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.<ref name="matiyasevich"/> 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.<ref>Template:Cite journal</ref> 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.<ref name="matiyasevich">Template:Cite book</ref>
Other examples of prime-generating formulas come from Mills' theorem and a theorem of Wright. These assert that there are real constants <math>A>1</math> and <math>\mu</math> such that
- <math>\left \lfloor A^{3^n}\right \rfloor \text{ and } \left \lfloor 2^{\cdots^{2^{2^\mu}}} \right \rfloor</math>
are prime for any natural number Template:Tmath in the first formula, and any number of exponents in the second formula.<ref>Template:Cite journal</ref> Here <math>\lfloor {}\cdot{} \rfloor</math> 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 Template:Tmath or <math>\mu.</math><ref name="matiyasevich"/>
Open questionsEdit
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.<ref>Template:Harvnb, p. vii.</ref> One of them is Goldbach's conjecture, which asserts that every even integer Template:Tmath greater than Template:Tmath can be written as a sum of two primes.<ref>Template:Harvnb, C1 Goldbach's conjecture, pp. 105–107.</ref> Template:As of, this conjecture has been verified for all numbers up to <math>n=4\cdot 10^{18}.</math><ref>Template:Cite journal</ref> 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.<ref>Template:Harvnb, 3.1 Structure and randomness in the prime numbers, pp. 239–247. See especially p. 239.</ref> 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).<ref>Template:Harvnb, p. 159.</ref> Also, any even integer greater than 10 can be written as the sum of six primes.<ref>Template:Cite journal</ref> The branch of number theory studying such questions is called additive number theory.<ref>Template:Cite book</ref>
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 <math>n!+2,n!+3,\dots,n!+n</math> consists of <math>n-1</math> composite numbers, for any natural number <math>n.</math><ref>Template:Harvnb, Theorem 2.14, p. 109. Template:Harvnb gives a similar argument using the primorial in place of the factorial.</ref> However, large prime gaps occur much earlier than this argument shows.<ref name="riesel-gaps"/> For example, the first prime gap of length 8 is between the primes 89 and 97,<ref>Template:Cite OEIS</ref> much smaller than <math>8!=40320.</math> 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 <math>k,</math> there are infinitely many pairs of consecutive primes that differ by <math>2k.</math><ref name="rib-gaps">Template:Harvnb, Gaps between primes, pp. 186–192.</ref> Andrica's conjecture,<ref name="rib-gaps"/> Brocard's conjecture,<ref name="rib-183">Template:Harvnb, p. 183.</ref> Legendre's conjecture,<ref name="chan">Template:Cite journal Note that Chan lists Legendre's conjecture as "Sierpinski's Postulate".</ref> and Oppermann's conjecture<ref name="rib-183"/> all suggest that the largest gaps between primes from 1 to Template:Tmath should be at most approximately <math>\sqrt{n},</math> a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at Template:Tmath.<ref name="rib-gaps"/> Prime gaps can be generalized to [[Prime k-tuple|prime Template:Tmath-tuples]], patterns in the differences among 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.<ref>Template:Harvnb, Prime Template:Tmath-tuples conjecture, pp. 201–202.</ref>
Analytic propertiesEdit
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 <math>1+\tfrac{1}{4}+\tfrac{1}{9}+\tfrac{1}{16}+\dots,</math> which today can be recognized as the value <math>\zeta(2)</math> 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 Template:Tmath.<ref>Template:Harvnb, Chapter 35, Estimating the Basel problem, pp. 205–208.</ref> The reciprocal of this number, Template:Tmath, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common).<ref>Template:Cite book</ref>
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 primes|formula for the Template:Tmath-th prime]] is known. Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
- <math>p(n) = a + bn</math>
with relatively prime integers Template:Tmath and Template:Tmath 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 Template:Tmath 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 theoremEdit
Euler's proof that there are infinitely many primes considers the sums of reciprocals of primes,
- <math>\frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p.</math>
Euler showed that, for any arbitrary real number Template:Tmath, there exists a prime Template:Tmath for which this sum is greater than Template:Tmath.<ref>Template:Harvnb, Section 1.6, Theorem 1.13</ref> 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 Template:Tmath. The growth rate of this sum is described more precisely by Mertens' second theorem.<ref>Template:Harvnb, Section 4.8, Theorem 4.12</ref> For comparison, the sum
- <math>\frac 1 {1^2} + \frac 1 {2^2} + \frac 1 {3^2} + \cdots + \frac 1 {n^2}</math>
does not grow to infinity as Template:Tmath 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.<ref name="mtb-invitation">Template:Cite book</ref> Brun's theorem states that the sum of the reciprocals of twin primes,
- <math> \left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1}Template:11 + \frac{1}Template:13} \right) + \cdots, </math>
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.<ref name="mtb-invitation"/>
Number of primes below a given boundEdit
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The prime-counting function <math>\pi(n)</math> is defined as the number of primes not greater than Template:Tmath.<ref>Template:Harvnb, p. 6.</ref> For example, Template:Tmath, since there are five primes less than or equal to 11. Methods such as the Meissel–Lehmer algorithm can compute exact values of <math>\pi(n)</math> faster than it would be possible to list each prime up to Template:Tmath.<ref>Template:Harvnb, Section 3.7, Counting primes, pp. 152–162.</ref> The prime number theorem states that <math>\pi(n)</math> is asymptotic to Template:Tmath, which is denoted as
- <math>\pi(n) \sim \frac{n}{\log n},</math>
and means that the ratio of <math>\pi(n)</math> to the right-hand fraction approaches 1 as Template:Tmath grows to infinity.<ref name="cranpom10">Template:Harvnb, p. 10.</ref> This implies that the likelihood that a randomly chosen number less than Template:Tmath is prime is (approximately) inversely proportional to the number of digits in Template:Tmath.<ref>Template:Cite book</ref> It also implies that the Template:Tmathth prime number is proportional to <math>n\log n</math><ref>Template:Harvnb, Section 4.6, Theorem 4.7</ref> and therefore that the average size of a prime gap is proportional to Template:Tmath.<ref name="riesel-gaps">Template:Harvnb, "Large gaps between consecutive primes", pp. 78–79.</ref> A more accurate estimate for <math>\pi(n)</math> is given by the offset logarithmic integral<ref name="cranpom10"/>
- <math>\pi(n)\sim \operatorname{Li}(n) = \int_2^n \frac{dt}{\log t}.</math>
Arithmetic progressionsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An arithmetic progression is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.<ref>Template:Cite book</ref> This difference is called the modulus of the progression.<ref>Template:Cite book</ref> For example,
- <math>3, 12, 21, 30, 39, ...,</math>
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
- <math>a, a+q, a+2q, a+3q, \dots</math>
can have more than one prime only when its remainder Template:Tmath and modulus Template:Tmath are relatively prime. If they are relatively prime, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes.<ref>Template:Harvnb, Theorem 1.1.5, p. 12.</ref> Template:Wide image The Green–Tao theorem shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.<ref name="neale-18-47"/><ref>Template:Cite journal</ref>
Prime values of quadratic polynomialsEdit
Euler noted that the function
- <math>n^2 - n + 41</math>
yields prime numbers for Template:Tmath, although composite numbers appear among its later values.<ref>Template:Cite book</ref><ref>The sequence of these primes, starting at <math>n=1</math> rather than Template:Tmath, is listed by Template:Cite book</ref> The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem.<ref>Template:Cite book</ref> 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.<ref name="guy-a1">Template:Cite book</ref>
The Ulam spiral<ref>Template:Cite journal</ref> 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.<ref name="guy-a1"/>
Zeta function and the Riemann hypothesisEdit
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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 <math>\zeta(s)</math> are located. This function is an analytic function on the complex numbers. For complex numbers Template:Tmath with real part greater than one it equals both an infinite sum over all integers, and an infinite product over the prime numbers,
- <math>\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}=\prod_{p \text{ prime}} \frac 1 {1-p^{-s}}.</math>
This equality between a sum and a product, discovered by Euler, is called an Euler product.<ref>Template:Cite book</ref> 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.<ref>Template:Cite book</ref> 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 Template:Tmath, but the sum would diverge (it is the harmonic series Template:Tmath) while the product would be finite, a contradiction.<ref>Template:Harvnb, pp. 191–193.</ref>
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.<ref>Template:Harvnb, Conjecture 2.7 (the Riemann hypothesis), p. 15.</ref> 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>Template:Harvnb, p. 7.</ref><ref name="bcrw18">Template:Harvnb, p. 18.</ref> although other more elementary proofs have been found.<ref>Template:Harvnb, Chapter 9, The prime number theorem, pp. 289–324.</ref> 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.<ref>Template:Cite journal See especially pp. 14–16.</ref> 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 Template:Tmath for intervals near a number Template:Tmath).<ref name="bcrw18"/>
Abstract algebraEdit
Modular arithmetic and finite fieldsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Modular arithmetic modifies usual arithmetic by only using the numbers Template:Tmath, for a natural number Template:Tmath called the modulus. Any other natural number can be mapped into this system by replacing it by its remainder after division by Template:Tmath.<ref>Template:Harvtxt, Proposition 5.3, p. 96.</ref> 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.<ref>Template:Cite book</ref> Equality of integers corresponds to congruence in modular arithmetic: Template:Tmath and Template:Tmath are congruent (written <math>x\equiv y</math> mod Template:Tmath) when they have the same remainder after division by Template:Tmath.<ref>Template:Harvnb, Theorem 3, p. 28.</ref> 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 7 as modulus, division by 3 is possible: Template:Tmath, because clearing denominators by multiplying both sides by 3 gives the valid formula Template:Tmath. However, with the composite modulus 6, division by 3 is impossible. There is no valid solution to <math>2/3\equiv x\bmod{6}</math>: clearing denominators by multiplying by 3 causes the left-hand side to become 2 while the right-hand side becomes either 0 or 3. 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.<ref>Template:Harvnb, pp. 27–28.</ref>
Several theorems about primes can be formulated using modular arithmetic. For instance, Fermat's little theorem states that if <math>a\not\equiv 0</math> (mod Template:Tmath), then <math>a^{p-1}\equiv 1</math> (mod Template:Tmath).<ref>Template:Harvnb, Fermat's little theorem and primitive roots modulo a prime, pp. 17–21.</ref> Summing this over all choices of Template:Tmath gives the equation
- <math>\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p,</math>
valid whenever Template:Tmath is prime. Giuga's conjecture says that this equation is also a sufficient condition for Template:Tmath to be prime.<ref>Template:Harvnb, The property of Giuga, pp. 21–22.</ref> Wilson's theorem says that an integer <math>p>1</math> is prime if and only if the factorial <math>(p-1)!</math> is congruent to <math>-1</math> mod Template:Tmath. For a composite number Template:Tmath this cannot hold, since one of its factors divides both Template:Mvar and Template:Tmath, and so <math>(n-1)!\equiv -1 \pmod{n}</math> is impossible.<ref>Template:Harvnb, The theorem of Wilson, p. 21.</ref>
p-adic numbersEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The [[p-adic order|Template:Tmath-adic order]] <math>\nu_p(n)</math> of an integer Template:Tmath is the number of copies of Template:Tmath in the prime factorization of Template:Tmath. The same concept can be extended from integers to rational numbers by defining the Template:Tmath-adic order of a fraction <math>m/n</math> to be Template:Tmath. The Template:Tmath-adic absolute value <math>|q|_p</math> of any rational number Template:Tmath is then defined as Template:Tmath. Multiplying an integer by its Template:Tmath-adic absolute value cancels out the factors of Template:Tmath 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 Template:Tmath-adic distance, the Template:Tmath-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 Template:Tmath. 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 Template:Tmath-adic distance can be extended to a different complete field, the [[p-adic number|Template:Tmath-adic numbers]].<ref name="childress"/><ref>Template:Cite book</ref>
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),<ref name="childress">Template:Cite book See also p. 64.</ref> and places (extensions to complete fields in which the given field is a dense set, also called completions).<ref>Template:Cite book Note however that some authors such as Template:Harvtxt instead use "place" to mean an equivalence class of norms.</ref> 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 Template:Tmath-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers.<ref name="childress"/> 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.<ref>Template:Cite book</ref>
Prime elements of a ringEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
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 Template:Tmath of a ring Template:Tmath is called prime if it is nonzero, has no multiplicative inverse (that is, it is not a unit), and satisfies the following requirement: whenever Template:Tmath divides the product <math>xy</math> of two elements of Template:Tmath, it also divides at least one of Template:Tmath or Template:Tmath. 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,
- <math>\{ \dots, -11, -7, -5, -3, -2, 2, 3, 5, 7, 11, \dots \}\, .</math>
In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for unique factorization domains.<ref>Template:Cite book</ref>
The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the Gaussian integers Template:Tmath, the ring of complex numbers of the form <math>a+bi</math> where Template:Tmath denotes the imaginary unit and Template:Tmath and Template:Tmath are arbitrary integers. Its prime elements are known as Gaussian primes. Not every number that is prime among the integers remains prime in the Gaussian integers; for instance, the number 2 can be written as a product of the two Gaussian primes <math>1+i</math> and Template:Tmath. Rational primes (the prime elements in the integers) congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.<ref>Template:Harvnb, Corollary 3.5.14, p. 133; Lemma 3.5.18, p. 136.</ref> This is a consequence of Fermat's theorem on sums of two squares, which states that an odd prime Template:Tmath is expressible as the sum of two squares, Template:Tmath, and therefore factorable as Template:Tmath, exactly when Template:Tmath is 1 mod 4.<ref>Template:Harvnb, Section 12.1, Sums of two squares, pp. 297–301.</ref>
Prime idealsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Not every ring is a unique factorization domain. For instance, in the ring of numbers <math>a+b\sqrt{-5}</math> (for integers Template:Tmath and Template:Tmath) the number <math>21</math> has two factorizations Template:Tmath, 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 principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, Template:Tmath, ... The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.<ref>Template:Cite book</ref>
The spectrum of a ring is a geometric space whose points are the prime ideals of the ring.<ref>Template:Cite book</ref> Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers.<ref>Template:Cite book</ref> Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers.<ref>Template:Harvnb, Section I.7, p. 38</ref> The question of how many integer prime numbers factor into a product of multiple prime ideals in an algebraic number field is addressed by Chebotarev's density theorem, which (when applied to the cyclotomic integers) has Dirichlet's theorem on primes in arithmetic progressions as a special case.<ref>Template:Cite journal</ref>
Group theoryEdit
In the theory of finite groups the Sylow theorems imply that, if a power of a prime number <math>p^n</math> divides the order of a group, then the group has a subgroup of order Template:Tmath. By Lagrange's theorem, any group of prime order is a cyclic group, and by Burnside's theorem any group whose order is divisible by only two primes is solvable.<ref>Template:Cite book For the Sylow theorems see p. 43; for Lagrange's theorem, see p. 12; for Burnside's theorem see p. 143.</ref>
Computational methodsEdit
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 mathematicsTemplate:Efn other than the use of prime numbered gear teeth to distribute wear evenly.<ref>Template:Cite book</ref> In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.<ref>Template:Cite book</ref>
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.<ref name="ent-7"/> 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 divisionEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The most basic method of checking the primality of a given integer Template:Tmath is called trial division. This method divides Template:Tmath by each integer from 2 up to the square root of Template:Tmath. Any such integer dividing Template:Tmath evenly establishes Template:Tmath as composite; otherwise it is prime. Integers larger than the square root do not need to be checked because, whenever Template:Tmath, one of the two factors Template:Tmath and Template:Tmath is less than or equal to the square root of Template:Tmath. Another optimization is to check only primes as factors in this range.<ref>Template:Cite book</ref> For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to Template:Tmath, 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.<ref>Template:Harvnb, p. 54</ref> 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.<ref name="p. 220">Template:Harvnb, p. 220.</ref>
SievesEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed.<ref>Template:Cite journal</ref> The oldest known method for generating a list of primes is called the sieve of Eratosthenes.<ref>Template:Cite book</ref> The animation shows an optimized variant of this method.<ref>Template:Cite book</ref> Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin.<ref>Template:Cite conference</ref> In advanced mathematics, sieve theory applies similar methods to other problems.<ref>Template:Cite book</ref>
Primality testing versus primality provingEdit
Some of the fastest modern tests for whether an arbitrary given number Template:Tmath is prime are probabilistic (or Monte Carlo) algorithms, meaning that they have a small random chance of producing an incorrect answer.<ref name="hromkovic">Template:Cite book</ref> For instance the Solovay–Strassen primality test on a given number Template:Tmath chooses a number Template:Tmath randomly from 2 through <math>p-2</math> and uses modular exponentiation to check whether <math>a^{(p-1)/2}\pm 1</math> is divisible by Template:Tmath.Template:Efn If so, it answers yes and otherwise it answers no. If Template:Tmath really is prime, it will always answer yes, but if Template:Tmath is composite then it answers yes with probability at most 1/2 and no with probability at least 1/2.<ref name="koblitz"/> If this test is repeated Template:Tmath times on the same number, the probability that a composite number could pass the test every time is at most Template:Tmath. Because this decreases exponentially with the number of tests, it provides high confidence (although not certainty) that a number that passes the repeated test is prime. On the other hand, if the test ever fails, then the number is certainly composite.<ref>Template:Cite book</ref> A composite number that passes such a test is called a pseudoprime.<ref name="koblitz">Template:Cite book</ref>
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,<ref name="tao-aks">Template:Cite book</ref> and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving.<ref name="hromkovic"/> When the elliptic curve method concludes that a number is prime, it provides primality certificate that can be verified quickly.<ref name="atkin-morain" /> The elliptic curve primality test is the fastest in practice of the guaranteed-correct primality tests, but its runtime analysis is based on heuristic arguments rather than rigorous proofs. The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice.<ref name="morain"/> These methods can be used to generate large random prime numbers, by generating and testing random numbers until finding one that is prime; when doing this, a faster probabilistic test can quickly eliminate most composite numbers before a guaranteed-correct algorithm is used to verify that the remaining numbers are prime.Template:Efn
The following table lists some of these tests. Their running time is given in terms of Template:Tmath, the number to be tested and, for probabilistic algorithms, the number Template:Tmath of tests performed. Moreover, <math>\varepsilon</math> is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that each time bound should be multiplied by a constant factor to convert it from dimensionless units to units of time; this factor depends on implementation details such as the type of computer used to run the algorithm, but not on the input parameters Template:Tmath and Template:Tmath.
| Test | Developed in | Type | Running time | Notes | References |
|---|---|---|---|---|---|
| AKS primality test | 2002 | deterministic | <math>O((\log n)^{6+\varepsilon})</math> | <ref name="tao-aks"/><ref>Template:Cite journal</ref> | |
| Elliptic curve primality proving | 1986 | Las Vegas | <math>O((\log n)^{4+\varepsilon})</math> heuristically | <ref name="morain">Template:Cite journal</ref> | |
| Baillie–PSW primality test | 1980 | Monte Carlo | <math>O((\log n)^{2+\varepsilon})</math> | <ref name="PSW">Template:Cite journal</ref><ref name="lpsp">Template:Cite journal</ref> | |
| Miller–Rabin primality test | 1980 | Monte Carlo | <math>O(k(\log n)^{2+\varepsilon})</math> | error probability <math>4^{-k}</math> | <ref name="monier">Template:Cite journal</ref> |
| Solovay–Strassen primality test | 1977 | Monte Carlo | <math>O(k(\log n)^{2+\varepsilon})</math> | error probability <math>2^{-k}</math> | <ref name="monier"/> |
Special-purpose algorithms and the largest known primeEdit
Template:Further 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 Lucas–Lehmer primality test can determine whether a Mersenne number (one less than a power of two) is prime, deterministically, in the same time as a single iteration of the Miller–Rabin test.<ref>Template:Cite book</ref> This is why since 1992 (Template:As of) the largest known prime has always been a Mersenne prime.<ref>Template:Harvnb, p. 41.</ref> It is conjectured that there are infinitely many Mersenne primes.<ref>For instance see Template:Harvnb, A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape Template:Tmath. pp. 13–21.</ref>
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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
| Type | Prime | Number of decimal digits | Date | Found by | ||
|---|---|---|---|---|---|---|
| Mersenne prime | 2136,279,841 − 1 | 41,024,320 | October 21, 2024<ref name="GIMPS-2024"/> | Luke Durant, Great Internet Mersenne Prime Search | ||
| Proth prime | 10,223 × 231,172,165 + 1 | 9,383,761 | citation | CitationClass=web
}}</ref> |
Péter Szabolcs, PrimeGrid<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> |
| factorial prime | 208,003! − 1 | 1,015,843 | July 2016 | citation | CitationClass=web
}}</ref> | |
| primorial primeTemplate:Efn | 1,098,133# − 1 | 476,311 | March 2012 | James P. Burt, PrimeGrid<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> | |
| twin primes | 2,996,863,034,895 × 21,290,000 ± 1 | 388,342 | September 2016 | Tom Greer, PrimeGrid<ref>{{#invoke:citation/CS1|citation | CitationClass=web
}}</ref> |
Integer factorizationEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Given a composite integer Template:Tmath, the task of providing one (or all) prime factors is referred to as factorization of Template:Tmath. It is significantly more difficult than primality testing,<ref>Template:Harvnb, p. 275.</ref> 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 Template:Tmath,<ref name="p. 220"/> and elliptic curve factorization can be effective when Template:Tmath has factors of moderate size.<ref>Template:Cite book</ref> Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve.<ref>Template:Cite journal</ref> Template:As of the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer.<ref>Template:Cite book</ref> However, current technology can only run this algorithm for very small numbers. Template:As of, the largest number that has been factored by a quantum computer running Shor's algorithm is 21.<ref>Template:Cite journal</ref>
Other computational applicationsEdit
Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (2048-bit primes are common).<ref>Template:Cite news</ref> RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers Template:Tmath and Template:Tmath than to calculate Template:Tmath and Template:Tmath (assumed coprime) if only the product <math>xy</math> is known.<ref name="ent-7"/> The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation (computing Template:Tmath), while the reverse operation (the discrete logarithm) is thought to be a hard problem.<ref>Template:Harvnb, Section 2.3, Diffie–Hellman key exchange, pp. 65–67.</ref>
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 [[k-independent hashing|Template:Tmath-independent hashing]] by using higher-degree polynomials, again modulo large primes.<ref>Template:Introduction to Algorithms For Template:Tmath-independent hashing see problem 11–4, p. 251. For the credit to Carter and Wegman, see the chapter notes, p. 252.</ref> 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.<ref>Template:Cite book See "Quadratic probing", p. 382, and exercise C–9.9, p. 415.</ref>
Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number. Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits.<ref>Template:Cite book</ref> Another checksum method, Adler-32, uses arithmetic modulo 65521, the largest prime number less than Template:Tmath.<ref>Template:Cite IETF</ref> Prime numbers are also used in pseudorandom number generators including linear congruential generators<ref>Template:Cite book</ref> and the Mersenne Twister.<ref>Template:Cite journal</ref>
Other applicationsEdit
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.<ref>Template:Cite journal</ref> 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.<ref>Template:Cite journal</ref>
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 prime field of a given field is its smallest subfield that contains both 0 and 1. It is either the field of rational numbers or a finite field with a prime number of elements, whence the name.<ref>Template:Cite book Section II.1, p. 90.</ref> Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.<ref>Template:Cite journal</ref> The prime decomposition of 3-manifolds is another example of this type.<ref>Template:Cite journal</ref>
Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics, and have been used metaphorically in the arts and literature. They have also been used in evolutionary biology to explain the life cycles of cicadas.
Constructible polygons and polygon partitionsEdit
Fermat primes are primes of the form
- <math>F_k = 2^{2^k}+1,</math>
with Template:Tmath a nonnegative integer.<ref>Template:Harvtxt also include Template:Tmath, which is not of this form.</ref> They are named after Pierre de Fermat, who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime,<ref name="kls">Template:Cite book</ref> but <math>F_5</math> is composite and so are all other Fermat numbers that have been verified as of 2017.<ref>Template:Cite journal</ref> A [[regular polygon|regular Template:Tmath-gon]] is constructible using straightedge and compass if and only if the odd prime factors of Template:Tmath (if any) are distinct Fermat primes.<ref name="kls"/> Likewise, a regular Template:Tmath-gon may be constructed using straightedge, compass, and an angle trisector if and only if the prime factors of [[regular polygon|Template:Tmath]] are any number of copies of 2 or 3 together with a (possibly empty) set of distinct Pierpont primes, primes of the form Template:Tmath.<ref>Template:Cite journal</ref>
It is possible to partition any convex polygon into Template:Tmath smaller convex polygons of equal area and equal perimeter, when Template:Tmath is a power of a prime number, but this is not known for other values of Template:Tmath.<ref>Template:Cite journal</ref>
Quantum mechanicsEdit
Beginning with the work of Hugh Montgomery and Freeman Dyson in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>Template:Cite journal</ref> Prime numbers are also significant in quantum information science, thanks to mathematical structures such as mutually unbiased bases and symmetric informationally complete positive-operator-valued measures.<ref>Template:Cite book</ref><ref>Template:Cite journal</ref>
BiologyEdit
The evolutionary strategy used by cicadas of the genus Magicicada makes use of prime numbers.<ref>Template:Cite journal</ref> These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles.<ref>Template:Cite journal</ref><ref>Template:Cite news</ref> In contrast, the multi-year periods between flowering in bamboo plants are hypothesized to be smooth numbers, having only small prime numbers in their factorizations.<ref>Template:Cite magazine</ref>
Arts and literatureEdit
Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–1950), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".<ref>Template:Cite book</ref>
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.<ref>Template:Cite book</ref> 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.<ref>Template:Cite news</ref> 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.<ref>Template:Cite news</ref>
NotesEdit
ReferencesEdit
External linksEdit
- Template:SpringerEOM
- Caldwell, Chris, The Prime Pages at primes.utm.edu.
- Template:In Our Time.
- "Teacher package: Prime numbers" from Plus, December 1, 2008, produced by the Millennium Mathematics Project at the University of Cambridge.
Generators and calculatorsEdit
- 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. Template:Webarchive.
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