View source for Prime number

{{Short description|Number divisible only by 1 or itself}}
[[File:Primes-vs-composites.svg|thumb|[[Composite number|Composite number]]s can be arranged into [[Rectangle|rectangle]]s but prime numbers cannot.|alt=Groups of two to twelve dots, showing that the composite numbers of dots (4, 6, 8, 9, 10, and 12) can be arranged into rectangles but prime numbers cannot]]

A '''prime number''' (or a '''prime''') is a [[Natural number|natural number]] greater than 1 that is not a [[Product (mathematics)|product]] of two smaller natural numbers. A natural number greater than 1 that is not prime is called a [[Composite number|composite number]]. For example, 5 is prime because the only ways of writing it as a product, {{nowrap|1 × 5}} or {{nowrap|5 × 1}}, involve 5 itself. However, 4 is composite because it is a product (2&nbsp;×&nbsp;2)<!-- {{nowrap}} avoided here so Navigation popups will display it --> in which both numbers are smaller than 4. Primes are central in [[Number theory|number theory]] because of the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]]: every natural number greater than 1 is either a prime itself or can be [[Factorization|factorized]] as a product of primes that is unique [[Up to|up to]] their order.

The property of being prime is called '''primality'''. A simple but slow [[Primality test|method of checking the primality]] of a given number <math>n</math>, called [[Trial division|trial division]], tests whether <math>n</math> is a multiple of any integer between 2 and <math>\sqrt{n}</math>. Faster algorithms include the [[Miller–Rabin primality test]], which is fast but has a small chance of error, and the [[AKS primality test]], which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as [[Mersenne number]]s. {{As of|2018|12}} the [[Largest known prime number|largest known prime number]] is a Mersenne prime with 24,862,048 [[Numerical digit|decimal digits]].<ref name="GIMPS-2018" />

There are [[Infinite set|infinitely many]] primes, as [[Euclid's theorem|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|prime number theorem]], proven at the end of the 19th century, which says that the [[Probability|probability]] of a randomly chosen large number being prime is inversely [[Proportionality (mathematics)|proportional]] to its number of digits, that is, to its [[Logarithm|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|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 number theory|analytic]] or [[Algebraic number theory|algebraic]] aspects of numbers. Primes are used in several routines in [[Engineering:Information technology|information technology]], such as [[Public-key cryptography|public-key cryptography]], which relies on the difficulty of [[Integer factorization|factoring]] large numbers into their prime factors. In [[Abstract algebra|abstract algebra]], objects that behave in a generalized way like prime numbers include [[Prime element|prime element]]s and prime ideals.

==Definition and examples==
{{main|List of prime numbers}}
A [[Natural number|natural number]] (1, 2, 3, 4, 5, 6, etc.) is called a ''prime number'' (or a ''prime'') if it is greater than 1 and cannot be written as the product of two smaller natural numbers. The numbers greater than 1 that are not prime are called [[Composite number|composite number]]s.<ref>{{cite book|title=The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965–1996|first=Anthony|last=Gardiner|publisher=Oxford University Press|year=1997|isbn=978-0-19-850105-3|page=[https://archive.org/details/mathematicalolym1997gard/page/26 26]|url=https://archive.org/details/mathematicalolym1997gard|url-access=registration}}</ref> In other words, <math>n</math> is prime if <math>n</math> items cannot be divided up into smaller equal-size groups of more than one item,<ref>{{cite book|title=Dyslexia, Dyscalculia and Mathematics: A practical guide|first=Anne|last=Henderson|edition=2nd|publisher=Routledge|year=2014|isbn=978-1-136-63662-2|page=62|url=https://books.google.com/books?id=uy-yGVRUilMC&pg=PA62}}</ref> or if it is not possible to arrange <math>n</math> dots into a rectangular grid that is more than one dot wide and more than one dot high.<ref>{{cite book|title=The Giant Golden Book of Mathematics: Exploring the World of Numbers and Space|url=https://archive.org/details/giantgoldenbooko00adle|url-access=registration |first=Irving|last=Adler|author-link=Irving Adler|publisher=Golden Press|year=1960|page=[https://archive.org/details/giantgoldenbooko00adle/page/16 16]|oclc=6975809}}</ref> For example, among the numbers 1 through 6, the numbers 2, 3, and 5 are the prime numbers,<ref>{{cite book | url = https://archive.org/details/barronsmathworkb00leff_0 | url-access = registration | page = [https://archive.org/details/barronsmathworkb00leff_0/page/360 360] | title = Math Workbook for the SAT I | first = Lawrence S. | last = Leff | publisher = Barron's Educational Series | year = 2000 | isbn = 978-0-7641-0768-9}}</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. {{nowrap|1=4 = 2 × 2}} and {{nowrap|1=6 = 2 × 3}} are both composite.

[[File:Prime number Cuisenaire rods 7.png|thumb|upright=1.2|Demonstration, with [[Cuisenaire rods|Cuisenaire rods]], that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly|alt=Demonstration, with Cuisenaire rods, that 7 is prime, because none of 2, 3, 4, 5, or 6 divide it evenly]]
The [[Divisor|divisor]]s of a natural number <math>n</math> are the natural numbers that divide <math>n</math> 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 [[Divisor|divisor]]s. Those two are 1 and the number itself. As 1 has only one divisor, itself, it is not prime by this definition.<ref>{{cite book | last1=Dudley | first1=Underwood |  title=Elementary number theory | publisher=W.H. Freeman and Co. | edition=2nd | isbn=978-0-7167-0076-0 | year=1978 | contribution=Section 2: Unique factorization | page=[https://archive.org/details/elementarynumber00dudl_0/page/10 10] | contribution-url=https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA10 | url=https://archive.org/details/elementarynumber00dudl_0/page/10 }}</ref> Yet another way to express the same thing is that a number <math>n</math> is prime if it is greater than one and if none of the numbers <math>2, 3, \dots, n-1</math> divides <math>n</math> evenly.<ref>{{cite book | last = Sierpiński | first = Wacław | author-link = Wacław Sierpiński | title = Elementary Theory of Numbers | edition = 2nd | volume = 31 | series = North-Holland Mathematical Library | publisher = Elsevier | year = 1988 | isbn = 978-0-08-096019-7 | page = 113 | url = https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA113 }}</ref>

The first 25 prime numbers (all the prime numbers less than 100) are:<!--Do not add 1 to this list. Its exclusion from the list is addressed in the "History of prime numbers" section below.--><ref name=ziegler>{{cite journal | last = Ziegler | first = Günter M. |  issue = 4 | journal = [[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]] | mr = 2039814 | pages = 414–416 | title = The great prime number record races | volume = 51 | year = 2004}}</ref>
: [[2|2]], [[3|3]], [[5|5]], [[7|7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[41 (number)|41]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]] {{OEIS|id=A000040}}.

No [[Even number|even number]] <math>n</math> greater than 2 is prime because any such number can be expressed as the product <math>2\times n/2</math>. Therefore, every prime number other than 2 is an [[Odd number|odd number]], and is called an ''odd prime''.<ref>{{cite book|title=Numbers and Geometry|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|publisher=Springer|year=1997|isbn=978-0-387-98289-2|page=9|url=https://books.google.com/books?id=4elkHwVS0eUC&pg=PA9}}</ref> Similarly, when written in the usual [[Decimal|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>{{cite book | last = Sierpiński | first = Wacław | author-link = Wacław Sierpiński | location = New York | mr = 0170843 | page = [https://archive.org/details/selectionproblem00sier/page/n37 40] | publisher = Macmillan | title = A Selection of Problems in the Theory of Numbers | url = https://archive.org/details/selectionproblem00sier | url-access = limited | year = 1964}}</ref>

The [[Set (mathematics)|set]] of all primes is sometimes denoted by <math>\mathbf{P}</math> (a boldface capital ''P'')<ref>{{cite book|title=Elementary Methods in Number Theory|volume=195|series=Graduate Texts in Mathematics|contribution=Notations and Conventions|contribution-url=https://books.google.com/books?id=sE7lBwAAQBAJ&pg=PP10|first=Melvyn B.|last=Nathanson|publisher=Springer|year=2000|isbn=978-0-387-22738-2|mr=1732941}}</ref> or by <math>\mathbb{P}</math> (a [[Blackboard bold|blackboard bold]] capital P).<ref>{{cite book|title=The Mathematics of Infinity: A Guide to Great Ideas|volume=111|series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts|first=Theodore G.|last=Faticoni|edition=2nd|publisher=John Wiley & Sons|year=2012|isbn=978-1-118-24382-4|page=44|url=https://books.google.com/books?id=I433i_ZGxRsC&pg=PA44}}</ref>

==History==
[[File:Rhind Mathematical Papyrus.jpg|thumb|The Rhind Mathematical Papyrus|alt=The Rhind Mathematical Papyrus]]

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 {{cite journal | last = Gillings | first = R.J. | doi = 10.1007/BF01307175 | journal = Archive for History of Exact Sciences | mr = 0497458 | pages = 291–298 | title = The recto of the Rhind Mathematical Papyrus. How did the ancient Egyptian scribe prepare it? | volume = 12 | issue = 4 | year = 1974| s2cid = 121046003 }}</ref> However, the earliest surviving records of the study of prime numbers come from the ancient Greek mathematians, who called them {{transl|grc|prōtos arithmòs}} ({{lang|grc|πρῶτος ἀριθμὸς}}). [[Biography:Euclid|Euclid]]'s ''[[Euclid's Elements|Elements]]'' (c. 300 BC) proves the infinitude of primes and the [[Fundamental theorem of arithmetic|fundamental theorem of arithmetic]], and shows how to construct a [[Perfect number|perfect number]] from a [[Mersenne prime]].<ref name="stillwell-2010-p40">{{cite book|title=Mathematics and Its History|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|edition=3rd |publisher=Springer |year=2010 |isbn=978-1-4419-6052-8 |page=40 |url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40}}</ref> Another Greek invention, the [[Sieve of Eratosthenes]], is still used to construct lists of {{nowrap|primes.<ref name="pomerance-sciam">{{cite journal |title=The Search for Prime Numbers|first=Carl|last=Pomerance|author-link=Carl Pomerance|journal=Scientific American|volume=247|issue=6|date=December 1982|pages=136–147|jstor=24966751|doi=10.1038/scientificamerican1282-136|bibcode=1982SciAm.247f.136P}}</ref><ref name="mollin">{{cite journal | last = Mollin | first = Richard A. | doi = 10.2307/3219180 | issue = 1 | journal = Mathematics Magazine | mr = 2107288 | pages = 18–29 | title = A brief history of factoring and primality testing B. C. (before computers) | volume = 75 | year = 2002| jstor = 3219180 }}</ref>}}

Around 1000 AD, the Islamic mathematician [[Biography:Ibn al-Haytham|Ibn al-Haytham]] (Alhazen) found [[Wilson's theorem]], characterizing the prime numbers as the numbers <math>n</math> that evenly divide <math>(n-1)!+1</math>. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham|mode=cs1}}</ref> Another Islamic mathematician, [[Biography:Ibn al-Banna' al-Marrakushi|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"/> [[Biography:Fibonacci|Fibonacci]] took the innovations from Islamic mathematics to Europe. His book ''[[Liber Abaci]]'' (1202) was the first to describe [[Trial division|trial division]] for testing primality, again using divisors only up to the square root.<ref name="mollin"/>

In 1640 [[Biography:Pierre de Fermat|Pierre de Fermat]] stated (without proof) [[Fermat's little theorem]] (later proved by [[Biography:Gottfried Wilhelm Leibniz|Leibniz]] and [[Biography:Leonhard Euler|Euler]]).<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA45 8. Fermat's Little Theorem (November 2003), p. 45]</ref> Fermat also investigated the primality of the [[Fermat number]]s <math>2^{2^n}+1</math>,<ref>{{cite book|title=How Euler Did Even More|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2014|isbn=978-0-88385-584-3|page=42|url=https://books.google.com/books?id=3c6iBQAAQBAJ&pg=PA42}}</ref> and [[Biography:Marin Mersenne|Marin Mersenne]] studied the [[Mersenne prime]]s, prime numbers of the form <math>2^p-1</math> with <math>p</math> itself a prime.<ref>{{cite book|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|publisher=Academic Press|year=2002|isbn=978-0-12-421171-1|page=369|url=https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA369}}</ref> [[Biography:Christian Goldbach|Christian Goldbach]] formulated [[Goldbach's conjecture]], that every even number is the sum of two primes, in a 1742 letter to Euler.<ref>{{cite book|title=Goldbach Conjecture|edition=2nd|volume=4|series=Series In Pure Mathematics|first=Wang|last=Yuan|publisher=World Scientific|year=2002|isbn=978-981-4487-52-8|page=21|url=https://books.google.com/books?id=g4jVCgAAQBAJ&pg=PA21}}</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|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 <math>\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{5}+\tfrac{1}{7}+\tfrac{1}{11}+\cdots</math>.<ref>{{cite book|title=The Development of Prime Number Theory: From Euclid to Hardy and Littlewood|series=Springer Monographs in Mathematics|first=Wladyslaw|last=Narkiewicz|publisher=Springer|year=2000|isbn=978-3-540-66289-1|page=11|contribution=1.2 Sum of Reciprocals of Primes|contribution-url=https://books.google.com/books?id=VVr3EuiHU0YC&pg=PA11}}</ref> At the start of the 19th century, Legendre and Gauss conjectured that as <math>x</math> tends to infinity, the number of primes up to <math>x</math> is [[Asymptotic analysis|asymptotic]] to <math>x/\log x</math>, where <math>\log x</math> is the [[Natural logarithm|natural logarithm]] of <math>x</math>. 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 <math>n</math> and <math>2n</math>, proved in 1852 by [[Biography:Pafnuty Chebyshev|Pafnuty Chebyshev]].<ref>{{cite journal|first=P. |last=Tchebychev |title=Mémoire sur les nombres premiers. |journal=Journal de mathématiques pures et appliquées |series=Série 1 |year=1852 |pages=366–390 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf |language=fr}}. (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 [[Biography:Bernhard Riemann|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 [[Biography:Jacques Hadamard|Hadamard]] and de la Vallée Poussin, and the result is now known as the [[Prime number theorem|prime number theorem]].<ref>{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | editor1-last = Bambah | editor1-first = R.P. | editor2-last = Dumir | editor2-first = V.C. | editor3-last = Hans-Gill | editor3-first = R.J. | contribution = A centennial history of the prime number theorem | location = Basel | mr = 1764793 | pages = 1–14 | publisher = Birkhäuser | series = Trends in Mathematics | title = Number Theory | contribution-url = https://books.google.com/books?id=aiDyBwAAQBAJ&pg=PA1 | year = 2000}}</ref> Another important 19th century result was [[Dirichlet's theorem on arithmetic progressions]], that certain [[Arithmetic progression|arithmetic progression]]s contain infinitely many primes.<ref>{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | contribution = 7. Dirichlet's Theorem on Primes in Arithmetical Progressions | contribution-url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA146 | location = New York; Heidelberg | mr = 0434929 | pages = 146–156 | publisher = Springer-Verlag | title = Introduction to Analytic Number Theory | year = 1976 }}</ref>

Many mathematicians have worked on [[Primality test|primality test]]s for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877),<ref>{{cite book|title=A History of Algorithms: From the Pebble to the Microchip|first=Jean-Luc|last=Chabert|publisher=Springer|year=2012|isbn=978-3-642-18192-4|page=261|url=https://books.google.com/books?id=XcDqCAAAQBAJ&pg=PA261}}</ref> [[Proth's theorem]] (c. 1878),<ref>{{cite book|title=Elementary Number Theory and Its Applications|first=Kenneth H.|last=Rosen|edition=4th|publisher=Addison-Wesley|year=2000|isbn=978-0-201-87073-2|contribution=Theorem 9.20. Proth's Primality Test|page=342}}</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 [[Computer|computer]]s.{{efn|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.<ref>{{cite book|title=The Once and Future Turing|first1=S. Barry|last1=Cooper|first2=Andrew|last2=Hodges|publisher=Cambridge University Press|year=2016|isbn=978-1-107-01083-3|pages=37–38|url=https://books.google.com/books?id=h12cCwAAQBAJ&pg=PA37}}</ref>}} The search for ever larger primes has generated interest outside mathematical circles, through the [[Great Internet Mersenne Prime Search]] and other [[Distributed computing|distributed computing]] projects.<ref name=ziegler/><ref>{{harvnb|Rosen|2000}}, p. 245.</ref> The idea that prime numbers had few applications outside of [[Pure mathematics|pure mathematics]]{{efn|name="pure"|For instance, Beiler writes that number theorist [[Biography:Ernst Kummer|Ernst Kummer]] loved his [[Ideal number|ideal number]]s, closely related to the primes, "because they had not soiled themselves with any practical applications",<ref>{{cite book|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|first=Albert H.|last=Beiler|year=1999|publisher=Dover|orig-year=1966|isbn=978-0-486-21096-4|page=2|url=https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA2|oclc=444171535}}</ref> and Katz writes that [[Biography:Edmund Landau|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|geometry]] that had already shown themselves to be useful.<ref>{{cite journal | last = Katz | first = Shaul | doi = 10.1017/S0269889704000092 | issue = 1–2 | journal = Science in Context | mr = 2089305 | pages = 199–234 | title = Berlin roots&nbsp;– Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem | volume = 17 | year = 2004| s2cid = 145575536 }}</ref>}} was shattered in the 1970s when [[Public-key cryptography|public-key cryptography]] and the [[RSA (cryptosystem)|RSA]] cryptosystem were invented, using prime numbers as their basis.<ref name="ent-7">{{cite book|title=Elementary Number Theory|series=Textbooks in mathematics|first1=James S.|last1=Kraft|first2=Lawrence C.|last2=Washington|publisher=CRC Press|year=2014|isbn=978-1-4987-0269-0|page=7|url=https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA7}}</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>{{cite book|title=Secret History: The Story of Cryptology|series=Discrete Mathematics and Its Applications|first=Craig P.|last=Bauer|publisher=CRC Press|year=2013|isbn=978-1-4665-6186-1|page=468|url=https://books.google.com/books?id=EBkEGAOlCDsC&pg=PA468}}</ref><ref>{{cite book|title=Old and New Unsolved Problems in Plane Geometry and Number Theory|volume=11|series=Dolciani mathematical expositions|first1=Victor|last1=Klee|first2=Stan|last2=Wagon|publisher=Cambridge University Press|year=1991|isbn=978-0-88385-315-3|page=224|url=https://books.google.com/books?id=tRdoIhHh3moC&pg=PA224}}</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 [[Biography:Yitang Zhang|Yitang Zhang]]'s 2013 proof that there exist infinitely many [[Prime gap|prime gap]]s of bounded size.<ref name="neale-18-47">{{harvnb|Neale|2017}}, pp. 18, 47.</ref>

===Primality of one===
Most early Greeks did not even consider 1 to be a number,<ref name="crxk-34">{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Reddick | first2 = Angela | last3 = Xiong | first3 = Yeng | last4 = Keller | first4 = Wilfrid | issue = 9 | journal = [[Journal of Integer Sequences]] | mr = 3005523 | page = Article 12.9.8 | title = The history of the primality of one: a selection of sources | url = https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html | volume = 15 | year = 2012 }} 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>{{cite book|title=Speusippus of Athens: A Critical Study With a Collection of the Related Texts and Commentary|volume=39|series=Philosophia Antiqua : A Series of Monographs on Ancient Philosophy|first=Leonardo|last=Tarán|publisher=Brill|year=1981|isbn=978-90-04-06505-5|pages=35–38|url=https://books.google.com/books?id=cUPXqSb7V1wC&pg=PA35}}</ref> so they could not consider its primality. A few scholars in the Greek and later Roman tradition, including [[Biography:Nicomachus|Nicomachus]], [[Biography:Iamblichus|Iamblichus]], [[Biography:Boethius|Boethius]], and [[Biography:Cassiodorus|Cassiodorus]] also considered the prime numbers to be a subdivision of the odd numbers, so they did not consider 2 to be prime either. However, Euclid and a majority of the other Greek mathematicians considered 2 as prime. The medieval Islamic mathematicians largely followed the Greeks in viewing 1 as not being a number.<ref name="crxk-34"/> By the Middle Ages and Renaissance, mathematicians began treating 1 as a number, and some of them included it as the first prime number.<ref>{{harvnb|Caldwell|Reddick|Xiong|Keller|2012}}, pp. 7–13. See in particular the entries for Stevin, Brancker, Wallis, and Prestet.</ref> In the mid-18th century [[Biography:Christian Goldbach|Christian Goldbach]] listed 1 as prime in his correspondence with [[Biography:Leonhard Euler|Leonhard Euler]]; however, Euler himself did not consider 1 to be prime.<ref>{{harvnb|Caldwell|Reddick|Xiong|Keller|2012}}, p. 15.</ref> In the 19th century many mathematicians still considered 1 to be prime,<ref name="cx"/> and lists of primes that included 1 continued to be published as recently {{nowrap|as 1956.<ref>{{cite book | last=Riesel | first=Hans |  title=Prime Numbers and Computer Methods for Factorization | publisher=Birkhäuser | location=Basel, Switzerland | isbn=978-0-8176-3743-9 | year=1994|page=36|edition=2nd|mr=1292250|url=https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA36 | doi=10.1007/978-1-4612-0251-6 }}</ref><ref name="cg-bon-129-130">{{cite book | last1=Conway | first1=John Horton |  last2=Guy | first2=Richard K. |  title=The Book of Numbers | url=https://archive.org/details/bookofnumbers0000conw | url-access=registration | publisher=Copernicus | location=New York | isbn=978-0-387-97993-9 | year=1996 | pages = [https://archive.org/details/bookofnumbers0000conw/page/129 129–130] | mr=1411676 | doi=10.1007/978-1-4612-4072-3 }}</ref>}}

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 {{nowrap|of 1.<ref name="cx">{{cite journal | last1 = Caldwell | first1 = Chris K. | last2 = Xiong | first2 = Yeng | issue = 9 | journal = [[Journal of Integer Sequences]] | mr = 3005530 | page = Article 12.9.7 | title = What is the smallest prime? | url = https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf | volume = 15 | year = 2012}}</ref>}} Similarly, the [[Sieve of Eratosthenes|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 {{harvnb|Sierpiński|1988}}, [https://books.google.com/books?id=ktCZ2MvgN3MC&pg=PA245 p. 245]. For the sum of divisors, see {{cite book|title=How Euler Did It|series=MAA Spectrum|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2007|isbn=978-0-88385-563-8|page=59|url=https://books.google.com/books?id=sohHs7ExOsYC&pg=PA59}}</ref> 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 (ring theory)|unit]]".<ref name="cx"/>

==Elementary properties==
===Unique factorization===
{{Main|Fundamental theorem of arithmetic}}

Writing a number as a product of prime numbers is called a ''prime factorization'' of the number. For example:

:<math>\begin{align}
   34866 &= 2\times 3\times 3\times 13 \times 149\\
   &=2\times 3^2\times 13 \times 149.
 \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>3.</math> When a prime occurs multiple times, [[Exponentiation|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>3^2</math> denotes the [[Square (algebra)|square]] or second power of <math>3.</math>

The central importance of prime numbers to number theory and mathematics in general stems from the ''fundamental theorem of arithmetic''.<ref>{{cite book|title=The Nature of Mathematics

Some proofs of the uniqueness of prime factorizations are based on [[Euclid's lemma]]: If <math>p</math> is a prime number and <math>p</math> divides a product <math>ab</math> of integers <math>a</math> and <math>b,</math> then <math>p</math> divides <math>a</math> or <math>p</math> divides <math>b</math> (or both).<ref>{{harvnb|Dudley|1978}}, [https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA15 Section 2, Lemma 5, p. 15]; {{cite book|title=Mathematics for the Curious|first=Peter M.|last=Higgins|publisher=Oxford University Press |year=1998 |isbn=978-0-19-150050-3 |pages=77–78|url=https://books.google.com/books?id=LeYH8P8S9oQC&pg=PA77}}</ref> Conversely, if a number <math>p</math> has the property that when it divides a product it always divides at least one factor of the product, then <math>p</math> must be prime.<ref>{{cite book|title=A First Course in Abstract Algebra|first=Joseph J.|last=Rotman|edition=2nd|publisher=Prentice Hall|year=2000|isbn=978-0-13-011584-3|at=Problem 1.40, p. 56}}</ref>

===Infinitude===
{{Main|Euclid's theorem}}
There are infinitely many prime numbers. Another way of saying this is that the sequence

: 2, 3, 5, 7, 11, 13, ...

of prime numbers never ends. This statement is referred to as ''Euclid's theorem'' in honor of the ancient Greek mathematician [[Biography:Euclid|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 [[Mathematical analysis|analytical]] proof by Euler, [[Biography:Christian Goldbach|Goldbach's]] [[Fermat number#Basic properties|proof]] based on [[Fermat number]]s,<ref>[http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0722.pdf Letter] in [[Social:Latin|Latin]] from Goldbach to Euler, July 1730.</ref> Furstenberg's [[Furstenberg's proof of the infinitude of primes|proof using general topology]],<ref>{{Cite journal | last1=Furstenberg | first1=Harry |  title=On the infinitude of primes | doi=10.2307/2307043 | year=1955 | journal=[[American Mathematical Monthly]]   | volume=62 | mr=0068566 | issue=5 | pages=353 | jstor=2307043 }}
</ref> and [[Biography:Ernst Kummer|Kummer's]] elegant proof.<ref>{{cite book | last1=Ribenboim | first1=Paulo |  title=The little book of bigger primes | publisher=Springer-Verlag | location=Berlin; New York | isbn=978-0-387-20169-6 | year=2004|page=4 | url=https://books.google.com/books?id=SvnTBwAAQBAJ&pg=PA5 }}</ref>

[[Euclid's theorem|Euclid's proof]]<ref>[[Euclid's Elements|Euclid's ''Elements'']], Book IX, Proposition 20. See [http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html David Joyce's English translation of Euclid's proof] or {{cite book|url=https://babel.hathitrust.org/cgi/pt?id=umn.31951000084215o;view=1up;seq=95|title=The Elements of Euclid, With Dissertations|last=Williamson|first=James|publisher=Clarendon Press|year=1782|location=Oxford|page=63|oclc=642232959}}</ref> shows that every [[Finite set|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, <math>N</math> has a prime factorization

: <math> N =  p'_1\cdot p'_2\cdots p'_m </math>

with one or more prime factors. <math>N</math> is evenly divisible by each of these factors, but <math>N</math> has a remainder of one when divided by any of the prime numbers in the given list, so none of the prime factors of <math>N</math> 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 number]]s.<ref>{{cite book|title=Computational Recreations in Mathematica|first=Ilan|last=Vardi|publisher=Addison-Wesley|year=1991|isbn=978-0-201-52989-0|pages=82–89}}</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 primes===
{{main|Formula for primes}}

There is no known efficient formula for primes. For example, there is no non-constant [[Polynomial|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>{{cite journal | last = Mackinnon | first = Nick | date = June 1987 | doi = 10.2307/3616496 | issue = 456 | pages = 113–114 | journal = The Mathematical Gazette | title = Prime number formulae | volume = 71| jstor = 3616496 | s2cid = 171537609 }}</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">{{cite book | last = Matiyasevich | first = Yuri V. |  year=1999 | chapter = Formulas for prime numbers | chapter-url=https://books.google.com/books?id=oLKlk5o6WroC&pg=PA13 | editor1-first=Serge | editor1-last = Tabachnikov |  title = Kvant Selecta: Algebra and Analysis | volume = II | publisher = American Mathematical Society | isbn = 978-0-8218-1915-9 | pages=13–24}}</ref>

Other examples of prime-generating formulas come from Mills' theorem and a theorem of [[Biography:E. M. Wright|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 <math>n</math> in the first formula, and any number of exponents in the second formula.<ref>{{cite journal |first=E.M. |last= Wright | title=A prime-representing function |journal=[[American Mathematical Monthly]] |volume=58 |issue=9 |year=1951 |pages=616–618 |jstor=2306356 |doi= 10.2307/2306356}}</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 <math>A</math> or <math>\mu.</math><ref name="matiyasevich"/>

===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.<ref>{{harvnb|Guy|2013}}, [https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PR7 p. vii].</ref> One of them is [[Goldbach's conjecture]], which asserts that every even integer <math>n</math> greater than 2 can be written as a sum of two primes.<ref>{{harvnb|Guy|2013}}, [https://books.google.com/books?id=EbLzBwAAQBAJ&pg=PA105 C1 Goldbach's conjecture, pp. 105–107].</ref> {{As of|2014}}, this conjecture has been verified for all numbers up to <math>n=4\cdot 10^{18}.</math><ref>{{cite journal | last1 = Oliveira e Silva | first1 = Tomás | last2 = Herzog | first2 = Siegfried | last3 = Pardi | first3 = Silvio | doi = 10.1090/S0025-5718-2013-02787-1 | issue = 288 | journal = [[Mathematics of Computation]] | mr = 3194140 | pages = 2033–2060 | title = Empirical verification of the even Goldbach conjecture and computation of prime gaps up to <math>4\cdot10^{18}</math> | volume = 83 | year = 2014| doi-access = free }}</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>{{harvnb|Tao|2009}}, [https://books.google.com/books?id=NxnVAwAAQBAJ&pg=PA239 3.1 Structure and randomness in the prime numbers, pp. 239–247]. See especially p.&nbsp;239.</ref> [[Chen's theorem]] says that every sufficiently large even number can be expressed as the sum of a prime and a [[Semiprime|semiprime]] (the product of two primes).<ref>{{harvnb|Guy|2013}}, p. 159.</ref> Also, any even integer greater than 10 can be written as the sum of six primes.<ref>{{cite journal | last = Ramaré | first = Olivier | issue = 4 | journal = Annali della Scuola Normale Superiore di Pisa | mr = 1375315 | pages = 645–706 | title = On Šnirel'man's constant | url = https://www.numdam.org/item?id=ASNSP_1995_4_22_4_645_0 | volume = 22 | year = 1995 | access-date = 2018-01-23 | archive-date = 2022-02-09 | archive-url = https://web.archive.org/web/20220209175544/http://www.numdam.org/item/?id=ASNSP_1995_4_22_4_645_0 | url-status = dead }}</ref> The branch of number theory studying such questions is called [[Additive number theory|additive number theory]].<ref>{{cite book | last = Rassias | first = Michael Th. | doi = 10.1007/978-3-319-57914-6 | isbn = 978-3-319-57912-2 | location = Cham | mr = 3674356 | page = vii | publisher = Springer | title = Goldbach's Problem: Selected Topics | url = https://books.google.com/books?id=ibwpDwAAQBAJ&pg=PP6 | year = 2017}}</ref>

Another type of problem concerns [[Prime gap|prime gap]]s, 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>{{harvnb|Koshy|2002}}, [https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA109 Theorem 2.14, p. 109]. {{harvnb|Riesel|1994}} gives a similar argument using the [[Primorial|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>{{Cite OEIS|A100964|name=Smallest prime number that begins a prime gap of at least 2n}}</ref> much smaller than <math>8!=40320.</math> It is conjectured that there are infinitely many [[Twin prime|twin prime]]s, 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">{{harvnb|Ribenboim|2004}}, Gaps between primes, pp. 186–192.</ref>
[[Andrica's conjecture]],<ref name="rib-gaps"/> [[Brocard's conjecture]],<ref name="rib-183">{{harvnb|Ribenboim|2004}}, p. 183.</ref> [[Legendre's conjecture]],<ref name="chan">{{cite journal | last = Chan | first = Joel | title = Prime time! | journal = Math Horizons | volume = 3 | issue = 3 | date = February 1996 | pages = 23–25 | jstor = 25678057| doi = 10.1080/10724117.1996.11974965 }} 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 <math>1</math> to <math>n</math> 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 <math>O((\log n)^2).</math><ref name="rib-gaps"/> Prime gaps can be generalized to [[Prime k-tuple|prime <math>k</math>-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|heuristic]] that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.<ref>{{harvnb|Ribenboim|2004}}, Prime <math>k</math>-tuples conjecture, pp. 201–202.</ref>

==Analytic properties==
[[Analytic number theory]] studies number theory through the lens of [[Continuous function|continuous function]]s, [[Limit (mathematics)|limits]], [[Series (mathematics)|infinite series]], and the related mathematics of the infinite and [[Infinitesimal|infinitesimal]].

This area of study began with [[Biography:Leonhard Euler|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 <math>\zeta(2)=\pi^2/6</math>.<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA205 Chapter 35, Estimating the Basel problem, pp. 205–208].</ref>
The reciprocal of this number, <math>6/\pi^2</math>, is the limiting probability that two random numbers selected uniformly from a large range are [[Coprime integers|relatively prime]] (have no factors in common).<ref>{{cite book | last1 = Ogilvy | first1 = C.S. |  last2 = Anderson | first2 = J.T. | isbn = 978-0-486-25778-5 | pages = 29–35 | publisher = Dover Publications Inc. | title = Excursions in Number Theory | url = https://books.google.com/books?id=efbaDLlTXvMC&pg=PA29 | year = 1988}}</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|prime number theorem]], but no efficient [[Formula for primes|formula for the <math>n</math>-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 <math>a</math> and <math>b</math> 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 <math>b</math> 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 [[Multiplicative inverse|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|real number]] <math>x</math>, there exists a prime <math>p</math> for which this sum is bigger than <math>x</math>.<ref>{{harvnb|Apostol|1976}}, 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 <math>x</math>.
The growth rate of this sum is described more precisely by [[Mertens' theorems|Mertens' second theorem]].<ref>{{harvnb|Apostol|1976}}, 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 <math>n</math> 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">{{cite book|title=An Invitation to Modern Number Theory|first1=Steven J.|last1=Miller|first2=Ramin|last2=Takloo-Bighash|publisher=Princeton University Press|year=2006|isbn=978-0-691-12060-7|pages=43–44|url=https://books.google.com/books?id=kLz4z8iwKiwC&pg=PA43}}</ref> [[Brun's theorem]] states that the sum of the reciprocals of [[Twin prime|twin prime]]s,

:<math> \left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1}{{11}} + \frac{1}{{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 bound===
{{Main|Prime number theorem|Prime-counting function}}
[[File:Prime-counting relative error.svg|thumb|upright=1.6|The [[Approximation error|relative error]] of <math>\tfrac{n}{\log n}</math> and the logarithmic integral <math>\operatorname{Li}(n)</math> as approximations to the [[Prime-counting function|prime-counting function]]. Both relative errors decrease to zero as <math>n</math> grows, but the convergence to zero is much more rapid for the logarithmic integral.]]
The [[Prime-counting function|prime-counting function]] <math>\pi(n)</math> is defined as the number of primes not greater than <math>n</math>.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA6 p.&nbsp;6].</ref> For example, <math>\pi(11)=5</math>, 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 <math>n</math>.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA152 Section 3.7, Counting primes, pp. 152–162].</ref> The [[Prime number theorem|prime number theorem]] states that <math>\pi(n)</math> is asymptotic to <math>n/\log n</math>, 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 <math>n</math> grows to infinity.<ref name="cranpom10">{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA10 p.&nbsp;10].</ref> This implies that the likelihood that a randomly chosen number less than <math>n</math> is prime is (approximately) inversely proportional to the number of digits {{nowrap|in <math>n</math>.<ref>{{cite book|title=The Number Mysteries: A Mathematical Odyssey through Everyday Life|first=Marcus|last=du Sautoy|publisher=St. Martin's Press|year=2011|isbn=978-0-230-12028-0|pages=50–52|contribution=What are the odds that your telephone number is prime?|contribution-url=https://books.google.com/books?id=snaUbkIb8SEC&pg=PA50}}</ref>}}
It also implies that the <math>n</math>th prime number is proportional to <math>n\log n</math><ref>{{harvnb|Apostol|1976}}, Section 4.6, Theorem 4.7</ref>
and therefore that the average size of a prime gap is proportional to <math>\log n</math>.<ref name="riesel-gaps">{{harvnb|Riesel|1994}}, "[https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA78 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 progressions===
{{main|Dirichlet's theorem on arithmetic progressions|Green–Tao theorem}}
An [[Arithmetic progression|arithmetic progression]] is a finite or infinite sequence of numbers such that consecutive numbers in the sequence all have the same difference.<ref>{{cite book|title=Algebra|first1=I.M.|last1=Gelfand|first2=Alexander|last2=Shen|publisher=Springer|year=2003|isbn=978-0-8176-3677-7|page=37|url=https://books.google.com/books?id=Z9z7iliyFD0C&pg=PA37}}</ref> This difference is called the [[Modular arithmetic|modulus]] of the progression.<ref>{{cite book|title=Fundamental Number Theory with Applications|series=Discrete Mathematics and Its Applications|first=Richard A.|last=Mollin|publisher=CRC Press|year=1997|isbn=978-0-8493-3987-5|page=76|url=https://books.google.com/books?id=Fsaa3MUUQYkC&pg=PA76}}</ref> 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
:<math>a, a+q, a+2q, a+3q, \dots</math>
can have more than one prime only when its remainder <math>a</math> and modulus <math>q</math> are relatively prime. If they are relatively prime, [[Dirichlet's theorem on arithmetic progressions]] asserts that the progression contains infinitely many primes.<ref>{{harvnb|Crandall|Pomerance|2005}}, [https://books.google.com/books?id=ZXjHKPS1LEAC&pg=PA Theorem 1.1.5, p. 12].</ref>
{{Wide image|Prime numbers in arithmetic progression mod&nbsp;9 zoom in.png|815px|Primes in the arithmetic progressions modulo 9. Each row of the thin horizontal band shows one of the nine possible progressions mod 9, with prime numbers marked in red. The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.|alt=Prime numbers in arithmetic progression mod 9}}
The [[Green–Tao theorem]] shows that there are arbitrarily long finite arithmetic progressions consisting only of primes.<ref name="neale-18-47"/><ref>{{cite journal|first1=Ben|last1=Green|first2=Terence|last2=Tao|title=The primes contain arbitrarily long arithmetic progressions|journal=[[Annals of Mathematics]]|volume=167|issue=2|year=2008|pages=481–547|doi=10.4007/annals.2008.167.481|arxiv=math.NT/0404188|s2cid=1883951}}</ref>

===Prime values of quadratic polynomials===
[[File:Ulam 2.png|thumb|upright=1.1|The [[Ulam spiral|Ulam spiral]]. Prime numbers (red) cluster on some diagonals and not others. Prime values of <math>4n^2 - 2n + 41</math> are shown in blue.|alt=The Ulam spiral]]
Euler noted that the function
:<math>n^2 - n + 41</math>
yields prime numbers for <math>1\le n\le 40</math>, although composite numbers appear among its later values.<ref>{{cite book|title=Additive Theory of Prime Numbers|last1=Hua|first1=L.K.|publisher=American Mathematical Society|year=2009|isbn=978-0-8218-4942-2|series=Translations of Mathematical Monographs|volume=13|location=Providence, RI|pages=176–177|mr=0194404|oclc=824812353|orig-year=1965}}</ref><ref>The sequence of these primes, starting at <math>n=1</math> rather than <math>n=0</math>, is listed by {{cite book|title=103 curiosità matematiche: Teoria dei numeri, delle cifre e delle relazioni nella matematica contemporanea|language=it|first1=Paolo Pietro|last1=Lava|first2=Giorgio|last2=Balzarotti|publisher=Ulrico Hoepli Editore S.p.A.|year=2010|isbn=978-88-203-5804-4|page=133|contribution-url=https://books.google.com/books?id=YfsSAAAAQBAJ&pg=PA133|contribution=Chapter 33. Formule fortunate}}</ref> The search for an explanation for this phenomenon led to the deep [[Algebraic number theory|algebraic number theory]] of [[Heegner number]]s and the [[Class number problem|class number problem]].<ref>{{cite book|title=Single Digits: In Praise of Small Numbers|first=Marc|last=Chamberland|publisher=Princeton University Press|year=2015|isbn=978-1-4008-6569-7|contribution=The Heegner numbers|pages=213–215|contribution-url=https://books.google.com/books?id=n9iqBwAAQBAJ&pg=PA213}}</ref> The Hardy–Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer [[Coefficient|coefficient]]s
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">{{cite book|title=Unsolved Problems in Number Theory|series=Problem Books in Mathematics|edition=3rd|first=Richard|last=Guy|publisher=Springer|year=2013|isbn=978-0-387-26677-0|pages=7–10|contribution-url=https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA7|contribution=A1 Prime values of quadratic functions}}</ref>

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.<ref name="guy-a1"/>

===Zeta function and the Riemann hypothesis===
{{Main|Riemann hypothesis}}
[[File:Riemann zeta function absolute value.png|thumb|upright=1.5|Plot of the absolute values of the zeta function, showing some of its features|alt=Plot of the absolute values of the zeta function]]
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 [[Zero of a function|zeros]] of the [[Riemann zeta function]] <math>\zeta(s)</math> are located.
This function is an [[Analytic function|analytic function]] on the [[Complex number|complex number]]s. For complex numbers <math>s</math> with real part greater than one it equals both an [[Series (mathematics)|infinite sum]] over all integers, and an [[Infinite product|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>{{cite book | last = Patterson | first = S.J. | doi = 10.1017/CBO9780511623707 | isbn = 978-0-521-33535-5 | mr = 933558 | page = 1 | publisher = Cambridge University Press, Cambridge | series = Cambridge Studies in Advanced Mathematics | title = An introduction to the theory of the Riemann zeta-function | url = https://books.google.com/books?id=IdHLCgAAQBAJ&pg=PA1 | volume = 14 | year = 1988 }}</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>{{cite book | last1 = Borwein | first1 = Peter |  last2 = Choi | first2 = Stephen | last3 = Rooney | first3 = Brendan | last4 = Weirathmueller | first4 = Andrea | doi = 10.1007/978-0-387-72126-2 | isbn = 978-0-387-72125-5 | location = New York | mr = 2463715 | pages = 10–11 | publisher = Springer | series = CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC | title = The Riemann hypothesis: A resource for the afficionado and virtuoso alike | url = https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA10 | year = 2008 }}</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 <math>s=1</math>, but the sum would diverge (it is the [[Harmonic series (mathematics)|harmonic series]] <math>1+\tfrac{1}{2}+\tfrac{1}{3}+\dots</math>) while the product would be finite, a contradiction.<ref>{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA191 pp. 191–193].</ref>

The Riemann hypothesis states that the [[Zero of a function|zeros]] of the zeta-function are all either negative even numbers, or complex numbers with real part equal to 1/2.<ref>{{harvnb|Borwein|Choi|Rooney|Weirathmueller|2008}}, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA15 Conjecture 2.7 (the Riemann hypothesis), p. 15].</ref> The original proof of the [[Prime number theorem|prime number theorem]] was based on a weak form of this hypothesis, that there are no zeros with real part equal to 1,<ref>{{harvnb|Patterson|1988}}, p. 7.</ref><ref name="bcrw18">{{harvnb|Borwein|Choi|Rooney|Weirathmueller|2008}}, [https://books.google.com/books?id=Qm1aZA-UwX4C&pg=PA18 p. 18.]</ref> although other more elementary proofs have been found.<ref>{{harvnb|Nathanson|2000}}, [https://books.google.com/books?id=sE7lBwAAQBAJ&pg=PA289 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>{{cite journal | last = Zagier | first = Don |  doi = 10.1007/bf03351556 | issue = S2 | journal = [[The Mathematical Intelligencer]] | pages = 7–19 | title = The first 50 million prime numbers | volume = 1 | year = 1977| s2cid = 37866599 }} 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|asymptotic distribution]] of primes given by the prime number theorem will also hold over much shorter intervals (of length about the square root of <math>x</math> for intervals near a number <math>x</math>).<ref name="bcrw18"/>

==Abstract algebra==

===Modular arithmetic and finite fields===
{{Main|Modular arithmetic}}
Modular arithmetic modifies usual arithmetic by only using the numbers <math>\{0,1,2,\dots,n-1\}</math>, for a natural number <math>n</math> called the modulus.
Any other natural number can be mapped into this system by replacing it by its remainder after division by <math>n</math>.<ref>{{harvtxt|Kraft|Washington|2014}}, [https://books.google.com/books?id=VG9YBQAAQBAJ&pg=PA96 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>{{cite book|title=Algebra in Action: A Course in Groups, Rings, and Fields|volume=27|series=Pure and Applied Undergraduate Texts|first=Shahriar|last=Shahriari|publisher=American Mathematical Society|year=2017|isbn=978-1-4704-2849-5|pages=20–21|url=https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA20}}</ref> Equality of integers corresponds to ''congruence'' in modular arithmetic:
<math>x</math> and <math>y</math> are congruent (written <math>x\equiv y</math> mod <math>n</math>) when they have the same remainder after division by <math>n</math>.<ref>{{harvnb|Dudley|1978}}, [https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA28 Theorem 3, p. 28].</ref> However, in this system of numbers, [[Division (mathematics)|division]] by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number <math>7</math> as modulus, division by <math>3</math> is possible: <math>2/3\equiv 3\bmod{7}</math>, because [[Clearing denominators|clearing denominators]] by multiplying both sides by <math>3</math> gives the valid formula <math>2\equiv 9\bmod{7}</math>. However, with the composite modulus <math>6</math>, division by <math>3</math> is impossible. There is no valid solution to <math>2/3\equiv x\bmod{6}</math>: clearing denominators by multiplying by <math>3</math> causes the left-hand side to become <math>2</math> while the right-hand side becomes either <math>0</math> or <math>3</math>.
In the terminology of [[Abstract algebra|abstract algebra]], the ability to perform division means that modular arithmetic modulo a prime number forms a [[Field (mathematics)|field]] or, more specifically, a [[Finite field|finite field]], while other moduli only give a [[Ring (mathematics)|ring]] but not a field.<ref>{{harvnb|Shahriari|2017}}, [https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA27 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 <math>p</math>), then <math>a^{p-1}\equiv 1</math> (mod <math>p</math>).<ref>{{harvnb|Ribenboim|2004}}, Fermat's little theorem and primitive roots modulo a prime, pp. 17–21.</ref>
Summing this over all choices of <math>a</math> gives the equation
:<math>\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p,</math>
valid whenever <math>p</math> is prime.
Giuga's conjecture says that this equation is also a sufficient condition for <math>p</math> to be prime.<ref>{{harvnb|Ribenboim|2004}}, 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|factorial]] <math>(p-1)!</math> is congruent to <math>-1</math> mod <math>p</math>. For a composite {{nowrap|number <math>\;n = r\cdot s\; </math>}} this cannot hold, since one of its factors divides both {{mvar|n}} and <math>(n-1)!</math>, and so <math>(n-1)!\equiv -1 \pmod{n}</math> is impossible.<ref>{{harvnb|Ribenboim|2004}}, The theorem of Wilson, p. 21.</ref>

===''p''-adic numbers===
{{main|P-adic number}}
The [[P-adic order|<math>p</math>-adic order]] <math>\nu_p(n)</math> of an integer <math>n</math> is the number of copies of <math>p</math> in the prime factorization of <math>n</math>. The same concept can be extended from integers to rational numbers by defining the <math>p</math>-adic order of a fraction <math>m/n</math> to be <math>\nu_p(m)-\nu_p(n)</math>. The <math>p</math>-adic absolute value <math>|q|_p</math> of any rational number <math>q</math> is then defined as
<math>|q|_p=p^{-\nu_p(q)}</math>. Multiplying an integer by its <math>p</math>-adic absolute value cancels out the factors of <math>p</math> 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 <math>p</math>-adic distance, the <math>p</math>-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 <math>p</math>. 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|complete field]], the rational numbers with the <math>p</math>-adic distance can be extended to a different complete field, the [[P-adic number|<math>p</math>-adic numbers]].<ref name="childress"/><ref>{{cite book | last1 = Erickson | first1 = Marty | last2 = Vazzana | first2 = Anthony | last3 = Garth | first3 = David | edition = 2nd | isbn = 978-1-4987-1749-6 | mr = 3468748 | page = 200 | publisher = CRC Press | location = Boca Raton, FL | series = Textbooks in Mathematics | title = Introduction to Number Theory | url = https://books.google.com/books?id=QpLwCgAAQBAJ&pg=PA200 | year = 2016}}</ref>

This picture of an order, absolute value, and complete field derived from them can be generalized to [[Algebraic number field|algebraic number field]]s and their [[Valuation (algebra)|valuations]] (certain mappings from the [[Multiplicative group|multiplicative group]] of the field to a totally ordered additive group, also called orders), [[Absolute value (algebra)|absolute values]] (certain multiplicative mappings from the field to the real numbers, also called norms),<ref name="childress">{{cite book | last = Childress | first = Nancy | doi = 10.1007/978-0-387-72490-4 | isbn = 978-0-387-72489-8 | mr = 2462595 | pages = 8–11 | publisher = Springer, New York | series = Universitext | title = Class Field Theory | url = https://books.google.com/books?id=RYdy4PCJYosC&pg=PA8 | year = 2009 }} See also p. 64.</ref> and places (extensions to [[Complete field|complete field]]s in which the given field is a [[Dense set|dense set]], also called completions).<ref>{{cite book | last = Weil | first = André |  isbn = 978-3-540-58655-5 | mr = 1344916 | page = [https://archive.org/details/basicnumbertheor00weil_866/page/n56 43] | publisher = Springer-Verlag | location = Berlin | series = Classics in Mathematics | title = Basic Number Theory | url = https://archive.org/details/basicnumbertheor00weil_866 | url-access = limited | year = 1995}} Note however that some authors such as {{harvtxt|Childress|2009}} instead use "place" to mean an equivalence class of norms.</ref> The extension from the rational numbers to the [[Real number|real number]]s, for instance, is a place in which the distance between numbers is the usual [[Absolute value|absolute value]] of their difference. The corresponding mapping to an additive group would be the [[Logarithm|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 <math>p</math>-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>{{cite book | last = Koch | first = H. | doi = 10.1007/978-3-642-58095-6 | isbn = 978-3-540-63003-6 | mr = 1474965 | page = 136 | publisher = Springer-Verlag | location = Berlin | title = Algebraic Number Theory | url = https://books.google.com/books?id=wt1sCQAAQBAJ&pg=PA136 | year = 1997| citeseerx = 10.1.1.309.8812 }}</ref>

===Prime elements in rings===
{{Main|Prime element|Irreducible element}}
[[File:Gaussian primes.png|thumb|The [[Gaussian prime|Gaussian prime]]s with norm less than 500]]
A [[Commutative ring|commutative ring]] is an [[Algebraic structure|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 <math>p</math> of a ring <math>R</math> is called prime if it is nonzero, has no [[Multiplicative inverse|multiplicative inverse]] (that is, it is not a [[Unit (ring theory)|unit]]), and satisfies the following requirement: whenever <math>p</math> divides the product <math>xy</math> of two elements of <math>R</math>, it also divides at least one of <math>x</math> or <math>y</math>. 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 domain|unique factorization domain]]s.<ref>{{cite book | last = Lauritzen | first = Niels | doi = 10.1017/CBO9780511804229 | isbn = 978-0-521-53410-9 | location = Cambridge | mr = 2014325 | page = 127 | publisher = Cambridge University Press | title = Concrete Abstract Algebra: From numbers to Gröbner bases | url = https://books.google.com/books?id=BdAbcje-TZUC&pg=PA127 | year = 2003 }}</ref>

The fundamental theorem of arithmetic continues to hold (by definition) in unique factorization domains. An example of such a domain is the [[Gaussian integer]]s <math>\mathbb{Z}[i]</math>, the ring of [[Complex number|complex number]]s of the form <math>a+bi</math> where <math>i</math> denotes the [[Imaginary unit|imaginary unit]] and <math>a</math> and <math>b</math> are arbitrary integers. Its prime elements are known as [[Gaussian prime]]s. 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 <math>1-i</math>. 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>{{harvnb|Lauritzen|2003}}, 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 <math>p</math> is expressible as the sum of two squares, <math>p=x^2+y^2</math>, and therefore factorable as <math>p=(x+iy)(x-iy)</math>, exactly when <math>p</math> is 1 mod 4.<ref>{{harvnb|Kraft|Washington|2014}}, [https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA297 Section 12.1, Sums of two squares, pp. 297–301].</ref>

===Prime ideals===
Not every ring is a unique factorization domain. For instance, in the ring of numbers <math>a+b\sqrt{-5}</math> (for integers <math>a</math> and <math>b</math>) the number <math>21</math> has two factorizations <math>21=3\cdot7=(1+2\sqrt{-5})(1-2\sqrt{-5})</math>, 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 (ring theory)|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|principal ideal]] generated by a prime element is a prime ideal, are an important tool and object of study in [[Commutative algebra|commutative algebra]], [[Number theory|algebraic number theory]] and [[Algebraic geometry|algebraic geometry]]. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), ... The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a [[Noetherian ring|Noetherian]] [[Commutative ring|commutative ring]] as an intersection of [[Primary ideal|primary ideal]]s, which are the appropriate generalizations of [[Prime power|prime power]]s.<ref>{{cite book | last1=Eisenbud | first1=David |  title=Commutative Algebra | publisher=Springer-Verlag | location=Berlin; New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1| mr=1322960 | year=1995 | volume=150 | at=Section 3.3| doi=10.1007/978-1-4612-5350-1 }}</ref>

The [[Spectrum of a ring|spectrum of a ring]] is a geometric space whose points are the prime ideals of the ring.<ref>{{cite book | last = Shafarevich | first = Igor R. |  doi = 10.1007/978-3-642-38010-5 | edition = 3rd | isbn = 978-3-642-38009-9 | mr = 3100288 | publisher = Springer, Heidelberg | title = Basic Algebraic Geometry 2: Schemes and Complex Manifolds | year = 2013 | contribution = Definition of <math>\operatorname{Spec} A</math> | page = 5 | contribution-url = https://books.google.com/books?id=zDW8BAAAQBAJ&pg=PA5}}</ref> [[Arithmetic geometry]] also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or [[Splitting of prime ideals in Galois extensions|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|ring of integers]] of quadratic number fields can be used in proving [[Quadratic reciprocity|quadratic reciprocity]], a statement that concerns the existence of square roots modulo integer prime numbers.<ref>{{cite book | last = Neukirch | first = Jürgen | author-link = Jürgen Neukirch | doi = 10.1007/978-3-662-03983-0 | isbn = 978-3-540-65399-8 | location = Berlin | mr = 1697859 | publisher = Springer-Verlag | series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | title = Algebraic Number Theory | volume = 322 | year = 1999 | at = Section I.8, p. 50}}</ref>
Early attempts to prove [[Fermat's Last Theorem]] led to [[Biography:Ernst Kummer|Kummer]]'s introduction of [[Regular prime|regular prime]]s, integer prime numbers connected with the failure of unique factorization in the [[Cyclotomic field|cyclotomic integers]].<ref>{{harvnb|Neukirch|1999}}, 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>{{cite journal | last1 = Stevenhagen | first1 = P. | last2 = Lenstra | first2 = H.W. Jr. |  doi = 10.1007/BF03027290 | issue = 2 | journal = [[The Mathematical Intelligencer]] | mr = 1395088 | pages = 26–37 | title = Chebotarëv and his density theorem | volume = 18 | year = 1996| citeseerx = 10.1.1.116.9409 | s2cid = 14089091 }}</ref>

===Group theory===
In the theory of [[Finite group|finite group]]s 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 <math>p^n</math>. By [[Lagrange's theorem (group theory)|Lagrange's theorem]], any group of prime order is a [[Cyclic group|cyclic group]],
and by [[Burnside's theorem]] any group whose order is divisible by only two primes is [[Solvable group|solvable]].<ref>{{cite book|title=The Theory of Groups|series=Dover Books on Mathematics|first=Marshall|last=Hall|publisher=Courier Dover Publications|year=2018|isbn=978-0-486-81690-6|url=https://books.google.com/books?id=K8hEDwAAQBAJ}} For the Sylow theorems see p. 43; for Lagrange's theorem, see p. 12; for Burnside's theorem see p. 143.</ref>

==Computational methods==
[[File:Gears large.jpg|thumb|The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13.]]
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{{efn|name="pure"}} other than the use of prime numbered gear teeth to distribute wear evenly.<ref>{{cite book|title=How Round is Your Circle?: Where Engineering and Mathematics Meet|title-link=How Round Is Your Circle|first1=John|last1=Bryant|first2=Christopher J.|last2=Sangwin|publisher=Princeton University Press|year=2008|isbn=978-0-691-13118-4|at=[https://books.google.com/books?id=iIN_2WjBH1cC&pg=PA178 p. 178]}}</ref> In particular, number theorists such as {{wipe|United Kingdom}} mathematician [[Biography:G. H. Hardy|G. H. Hardy]] prided themselves on doing work that had absolutely no military significance.<ref>{{cite book|title=A Mathematician's Apology|last1=Hardy|first1=Godfrey Harold|publisher=Cambridge University Press|year=2012|isbn=978-0-521-42706-7|page=[https://books.google.com/books?id=EkY2im6xkVkC&pg=PA140 140]|oclc=922010634|quote=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.|orig-year=1940|title-link=A Mathematician's Apology}}</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|public-key cryptography]] algorithms.<ref name="ent-7"/>
These applications have led to significant study of [[Algorithm|algorithm]]s for computing with prime numbers, and in particular of [[Primality test|primality test]]ing, 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 [[Integer factorization|factorization]] algorithms.
Prime numbers are also used in computing for [[Checksum|checksum]]s, [[Hash table|hash table]]s, and [[Pseudorandom number generator|pseudorandom number generator]]s.

===Trial division===
{{main|Trial division}}
The most basic method of checking the primality of a given integer <math>n</math> is called ''[[Trial division|trial division]]''. This method divides <math>n</math> by each integer from 2 up to the [[Square root|square root]] of <math>n</math>. Any such integer dividing <math>n</math> evenly establishes <math>n</math> as composite; otherwise it is prime.
Integers larger than the square root do not need to be checked because, whenever <math>n = a\cdot b</math>, one of the two factors <math>a</math> and <math>b</math> is less than or equal to the [[Square root|square root]] of <math>n</math>. Another optimization is to check only primes as factors in this range.<ref>{{cite book|title=Primes and Programming|first=Peter|last=Giblin|publisher=Cambridge University Press|year=1993|isbn=978-0-521-40988-9|page=[https://archive.org/details/primesprogrammin0000gibl/page/39 39]|url=https://archive.org/details/primesprogrammin0000gibl|url-access=registration}}</ref>
For instance, to check whether 37 is prime, this method divides it by the primes in the range from 2 to <math>\sqrt{37}</math>, 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 [[Exponential growth|grows exponentially]] as a function of the number of digits of these integers.<ref>{{harvnb|Giblin|1993}}, [https://archive.org/details/primesprogrammin0000gibl/page/54 p.&nbsp;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">{{harvnb|Riesel|1994}}, [https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA220 p.&nbsp;220].</ref>

===Sieves===
[[File:Sieve of Eratosthenes animation.gif|frame|The [[Sieve of Eratosthenes|sieve of Eratosthenes]] starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc.) are marked as primes (magenta).|alt=Animation of the sieve of Eratosthenes]]

{{main|Sieve of Eratosthenes}}
Before computers, [[Mathematical table|mathematical table]]s listing all of the primes or prime factorizations up to a given limit were commonly printed.<ref>{{cite journal|first=Maarten|last=Bullynck|title=A history of factor tables with notes on the birth of number theory 1657–1817|journal=Revue d'Histoire des Mathématiques|year=2010|pages=133–216|volume=16|issue=2|url=https://hal-univ-paris8.archives-ouvertes.fr/hal-01103903/}}</ref> The oldest method for generating a list of primes is called the sieve of Eratosthenes.<ref>{{cite book|title=The Joy of Factoring|volume=68|series=Student mathematical library|first=Samuel S. Jr.|last=Wagstaff|publisher=American Mathematical Society|year=2013|isbn=978-1-4704-1048-3|page=191|url=https://books.google.com/books?id=rowCAQAAQBAJ&pg=PA191}}</ref> The animation shows an optimized variant of this method.<ref>{{cite book|title=Prime Numbers: A Computational Perspective|edition=2nd|first1=Richard|last1=Crandall|first2=Carl|last2=Pomerance|publisher=Springer|year=2005|isbn=978-0-387-25282-7|page=121|url=https://books.google.com/books?id=RbEz-_D7sAUC&pg=PA121}}</ref>
Another more asymptotically efficient sieving method for the same problem is the sieve of Atkin.<ref>{{cite conference | last1 = Farach-Colton | first1 = Martín |  last2 = Tsai | first2 = Meng-Tsung | editor1-last = Elbassioni | editor1-first = Khaled | editor2-last = Makino | editor2-first = Kazuhisa | arxiv = 1504.05240 | contribution = On the complexity of computing prime tables | doi = 10.1007/978-3-662-48971-0_57 | pages = 677–688 | publisher = Springer | series = Lecture Notes in Computer Science | title = Algorithms and Computation: 26th International Symposium, ISAAC 2015, Nagoya, Japan, December 9-11, 2015, Proceedings | volume = 9472 | year = 2015}}</ref> In advanced mathematics, sieve theory applies similar methods to other problems.<ref>{{cite book|title=Sieves in Number Theory|volume=43|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge)|first=George|last=Greaves|publisher=Springer|year=2013|isbn=978-3-662-04658-6|page=1|url=https://books.google.com/books?id=G0TtCAAAQBAJ&pg=PA1}}</ref>

===Primality testing versus primality proving===
Some of the fastest modern tests for whether an arbitrary given number <math>n</math> is prime are probabilistic (or [[Monte Carlo algorithm|Monte Carlo]]) algorithms, meaning that they have a small random chance of producing an incorrect answer.<ref name="hromkovic">{{cite book | last = Hromkovič | first = Juraj |  contribution = 5.5 Bibliographic Remarks | contribution-url = https://books.google.com/books?id=nkeqCAAAQBAJ&pg=PA383 | doi = 10.1007/978-3-662-04616-6 | isbn = 978-3-540-66860-2 | mr = 1843669 | pages = 383–385 | publisher = Springer-Verlag, Berlin | series = Texts in Theoretical Computer Science. An EATCS Series | title = Algorithmics for Hard Problems | year = 2001| s2cid = 31159492 }}</ref>
For instance the [[Solovay–Strassen primality test]] on a given number <math>p</math> chooses a number <math>a</math> randomly from <math>2</math> through <math>p-2</math> and uses [[Modular exponentiation|modular exponentiation]] to check
whether <math>a^{(p-1)/2}\pm 1</math> is divisible by <math>p</math>.{{efn|In this test, the <math>\pm 1</math> term is negative if <math>a</math> is a square modulo the given (supposed) prime <math>p</math>, and positive otherwise. More generally, for non-prime values of <math>p</math>, the <math>\pm 1</math> term is the (negated)
 [[Jacobi symbol]], which can be calculated using [[Quadratic reciprocity|quadratic reciprocity]].}} If so, it answers yes and otherwise it answers no. If <math>p</math> really is prime, it will always answer yes, but if <math>p</math> 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 <math>n</math> times on the same number,
the probability that a composite number could pass the test every time is at most <math>1/2^n</math>. 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>{{cite book | title = Fundamentals of Computer Security | first1 = Josef | last1 = Pieprzyk | first2 = Thomas | last2 = Hardjono | first3 = Jennifer | last3 = Seberry |  publisher = Springer | year = 2013 | isbn = 978-3-662-07324-7 | contribution = 2.3.9 Probabilistic Computations | pages = 51–52 | contribution-url = https://books.google.com/books?id=BG2rCAAAQBAJ&pg=PA51}}</ref>
A composite number that passes such a test is called a pseudoprime.<ref name="koblitz">{{cite book | last = Koblitz | first = Neal |  contribution = Chapter V. Primality and Factoring | doi = 10.1007/978-1-4684-0310-7_5 | isbn = 978-0-387-96576-5 | mr = 910297 | pages = 112–149 | publisher = Springer-Verlag, New York | series = Graduate Texts in Mathematics | title = A Course in Number Theory and Cryptography | volume = 114 | year = 1987}}</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 algorithm|deterministic]] (non-random) algorithms, such as the [[AKS primality test]],<ref name="tao-aks">{{cite book | last = Tao | first = Terence |  contribution = 1.11 The AKS primality test | contribution-url = https://terrytao.wordpress.com/2009/08/11/the-aks-primality-test/ | doi = 10.1090/gsm/117 | isbn = 978-0-8218-5280-4 | location = Providence, RI | mr = 2780010 | pages = 82–86 | publisher = American Mathematical Society | title = An epsilon of room, II: Pages from year three of a mathematical blog | volume = 117 | year = 2010| series = Graduate Studies in Mathematics }}</ref>
and randomized [[Las Vegas algorithm]]s 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 argument|heuristic argument]]s 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.{{efn|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.<ref name = "atkin-morain">{{cite journal | last1 = Atkin | first1 = A O.L. |  last2 = Morain | first2 = F. | doi =  10.1090/s0025-5718-1993-1199989-x| issue = 203 | journal = [[Mathematics of Computation]] | mr = 1199989 | pages = 29–68 | title = Elliptic curves and primality proving | volume = 61 | year = 1993| jstor = 2152935 | bibcode = 1993MaCom..61...29A |url = https://www.ams.org/mcom/1993-61-203/S0025-5718-1993-1199989-X/S0025-5718-1993-1199989-X.pdf | doi-access = free }}</ref>}}

The following table lists some of these tests. Their running time is given in terms of <math>n</math>, the number to be tested and, for probabilistic algorithms, the number <math>k</math> of tests performed. Moreover, <math>\varepsilon</math> is an arbitrarily small positive number, and log is the [[Logarithm|logarithm]] to an unspecified base. The [[Big O notation|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 <math>n</math> and <math>k</math>.

{| class="wikitable sortable"
|-
! Test
! Developed in
! Type
! Running time
! Notes
! class=unsortable| References
|-
| [[AKS primality test]]
| 2002
| deterministic
| <math>O((\log n)^{6+\varepsilon})</math>
|
|-
| Elliptic curve primality proving
| 1986
| Las Vegas
| <math>O((\log n)^{4+\varepsilon})</math> ''heuristically''
|
| <ref name="morain">{{cite journal | last = Morain | first = F. | doi = 10.1090/S0025-5718-06-01890-4 | issue = 257 | journal = [[Mathematics of Computation]] | mr = 2261033 | pages = 493–505 | title = Implementing the asymptotically fast version of the elliptic curve primality proving algorithm | volume = 76 | year = 2007| arxiv = math/0502097 | bibcode = 2007MaCom..76..493M | s2cid = 133193 }}</ref>
|-
| Baillie–PSW primality test
| 1980
| Monte Carlo
| <math>O((\log n)^{2+\varepsilon})</math>
|
|-
| [[Miller–Rabin primality test]]
| 1980
| Monte Carlo
| <math>O(k(\log n)^{2+\varepsilon})</math>
| error probability <math>4^{-k}</math>
| <ref name="monier">{{cite journal | last = Monier | first = Louis | doi = 10.1016/0304-3975(80)90007-9 | issue = 1 | journal = Theoretical Computer Science | mr = 582244 | pages = 97–108 | title = Evaluation and comparison of two efficient probabilistic primality testing algorithms | volume = 12 | year = 1980| doi-access = free }}</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 prime===
{{Further|List of prime numbers}}
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|power of two]]) is prime, deterministically,
in the same time as a single iteration of the Miller–Rabin test.<ref>{{cite book | last = Tao | first = Terence |  contribution = 1.7 The Lucas–Lehmer test for Mersenne primes | contribution-url = https://terrytao.wordpress.com/2008/10/02/the-lucas-lehmer-test-for-mersenne-primes/ | isbn = 978-0-8218-4883-8 | location = Providence, RI | mr = 2523047 | pages = 36–41 | publisher = American Mathematical Society | title = Poincaré's legacies, pages from year two of a mathematical blog. Part I | year = 2009 }}</ref> This is why since 1992 ({{as of|2018|12|lc=y}}) the largest ''known'' prime has always been a Mersenne prime.<ref>{{harvnb|Kraft|Washington|2014}}, [https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA41 p.&nbsp;41].</ref>
It is conjectured that there are infinitely many Mersenne primes.<ref>For instance see {{harvnb|Guy|2013}}, [https://books.google.com/books?id=1BnoBwAAQBAJ&pg=PA13 A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape <math>k\cdot 2^n+1</math>. 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|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>{{cite web | url= https://www.eff.org/press/archives/2009/10/14-0  | title= Record 12-Million-Digit Prime Number Nets $100,000 Prize | date= October 14, 2009 | publisher= Electronic Frontier Foundation | access-date= 2010-01-04 }}</ref> The [[Organization:Electronic Frontier Foundation|Electronic Frontier Foundation]] also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.<ref>{{cite web  | url= https://www.eff.org/awards/coop | title= EFF Cooperative Computing Awards | date= 2008-02-29| publisher= Electronic Frontier Foundation | access-date= 2010-01-04 }}</ref>

{| class="wikitable"
|-
! Type
! Prime
! Number of decimal digits
! Date
! Found by
|-
| [[Mersenne prime]]
| 2<sup>82,589,933</sup> − 1
| style="text-align:right;"| 24,862,048
| December 7, 2018<ref name="GIMPS-2018">{{cite web |title=GIMPS Project Discovers Largest Known Prime Number: 2<sup>82,589,933</sup>-1 |url=https://www.mersenne.org/primes/press/M82589933.html |date=21 December 2018 |work=Mersenne Research, Inc. |access-date=21 December 2018 }}</ref>
| Patrick Laroche, [[Great Internet Mersenne Prime Search]]
|-
| [[Proth prime]]
| 10,223 × 2<sup>31,172,165</sup> + 1
| style="text-align:right;"| 9,383,761
| October 31, 2016<ref>{{cite web|url=https://www.primegrid.com/download/SOB-31172165.pdf|title=PrimeGrid's Seventeen or Bust Subproject|access-date=2017-01-03}}</ref>
| Péter Szabolcs, [[PrimeGrid]]<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=3 |title=The Top Twenty: Largest Known Primes |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|-
| [[Factorial prime|factorial prime]]
| 208,003! − 1
| style="text-align:right;"| 1,015,843
| July 2016
| Sou Fukui<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=30 |title=The Top Twenty: Factorial |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|-
| [[Primorial prime|primorial prime]]{{efn|The [[Primorial|primorial]] function of <math>n</math>, denoted by <math>n\#</math>, yields the product of the prime numbers up to <math>n</math>, and a [[Primorial prime|primorial prime]] is a prime of one of the forms <math>n\#\pm 1</math>.<ref>{{harvnb|Ribenboim|2004}}, p. 4.</ref>}}
| 1,098,133# − 1
| style="text-align:right;"| 476,311
| March 2012
| James P. Burt, [[PrimeGrid]]<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=5 |title=The Top Twenty: Primorial |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|-
| [[Twin prime|twin prime]]s
| 2,996,863,034,895 × 2<sup>1,290,000</sup> ± 1
| style="text-align:right;"| 388,342
| September 2016
| Tom Greer, [[PrimeGrid]]<ref>{{cite web|first=Chris K.|last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=1 |title=The Top Twenty: Twin Primes |website=[[Prime Pages|The Prime Pages]] |access-date=2017-01-03}}</ref>
|}

===Integer factorization===
{{main|Integer factorization}}
Given a composite integer <math>n</math>, the task of providing one (or all) prime factors is referred to as ''factorization'' of <math>n</math>. It is significantly more difficult than primality testing,<ref>{{harvnb|Kraft|Washington|2014}}, [https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA275 p.&nbsp;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 <math>n</math>,<ref name="p. 220"/> and elliptic curve factorization can be effective when <math>n</math> has factors of moderate size.<ref>{{cite book|title=An Introduction to Mathematical Cryptography|series=Undergraduate Texts in Mathematics|first1=Jeffrey|last1=Hoffstein|first2=Jill|last2=Pipher|first3=Joseph H.|last3=Silverman|edition=2nd|publisher=Springer|year=2014|isbn=978-1-4939-1711-2|page=329|url=https://books.google.com/books?id=cbl_BAAAQBAJ&pg=PA329}}</ref> Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the [[Quadratic sieve|quadratic sieve]] and [[General number field sieve|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|special number field sieve]].<ref>{{cite journal | last = Pomerance | first = Carl |  issue = 12 | journal = [[Organization:Notices of the American Mathematical Society|Notices of the American Mathematical Society]] | mr = 1416721 | pages = 1473–1485 | title = A tale of two sieves | volume = 43 | year = 1996}}</ref> {{as of|2019|12}} the [[Integer factorization records|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>Emmanuel Thomé, [https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;fd743373.1912 “795-bit factoring and discrete logarithms,”] December 2, 2019.</ref>

[[Shor's algorithm]] can factor any integer in a polynomial number of steps on a quantum computer.<ref>{{cite book|title=Quantum Computing: A Gentle Introduction|first1=Eleanor G.|last1=Rieffel|first2=Wolfgang H.|last2=Polak|publisher=MIT Press|year=2011|isbn=978-0-262-01506-6|contribution=Chapter 8. Shor's Algorithm|pages=163–176|title-link= Quantum Computing: A Gentle Introduction |contribution-url=https://books.google.com/books?id=iYX6AQAAQBAJ&pg=PA163}}</ref> However, current technology can only run this algorithm for very small numbers. {{As of|2012|10}} the largest number that has been factored by a quantum computer running Shor's algorithm is 21.<ref>{{cite journal |last1=Martín-López |first1=Enrique  |first2=Anthony|last2=Laing|first3=Thomas|last3=Lawson  |first4=Roberto|last4=Alvarez  |first5=Xiao-Qi|last5=Zhou  |first6=Jeremy L.|last6=O'Brien |title=Experimental realization of Shor's quantum factoring algorithm using qubit recycling |journal=Nature Photonics |volume=6  |issue=11  |pages=773–776  |date=12 October 2012 |doi=10.1038/nphoton.2012.259 |arxiv = 1111.4147  |bibcode = 2012NaPho...6..773M |s2cid=46546101 }}</ref>

===Other computational applications===
Several [[Public-key cryptography|public-key cryptography]] algorithms, such as RSA and the [[Diffie–Hellman key exchange]], are based on large prime numbers (2048-[[Bit|bit]] primes are common).<ref>{{cite news|newspaper=The Register|url=https://www.theregister.co.uk/2016/10/09/crypto_needs_more_transparency_researchers_warn/|title=Crypto needs more transparency, researchers warn|first=Richard|last=Chirgwin|date=October 9, 2016}}</ref> RSA relies on the assumption that it is much easier (that is, more efficient) to perform the multiplication of two (large) numbers <math>x</math> and <math>y</math> than to calculate <math>x</math> and <math>y</math> (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|modular exponentiation]] (computing <math>a^b\bmod{c}</math>), while the reverse operation (the [[Discrete logarithm|discrete logarithm]]) is thought to be a hard problem.<ref>{{harvnb|Hoffstein|Pipher|Silverman|2014}}, Section 2.3, Diffie–Hellman key exchange, pp. 65–67.</ref>

Prime numbers are frequently used for [[Hash table|hash table]]s. For instance the original method of Carter and Wegman for [[Universal hashing|universal hashing]] was based on computing [[Hash function|hash function]]s by choosing random [[Linear function|linear function]]s modulo large prime numbers. Carter and Wegman generalized this method to [[K-independent hashing|<math>k</math>-independent hashing]] by using higher-degree polynomials, again modulo large primes.<ref>{{Introduction to Algorithms|edition=2|chapter=11.3 Universal hashing|pages=232–236}} For <math>k</math>-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|quadratic probing]] based hash tables to ensure that the probe sequence covers the whole table.<ref>{{cite book|title=Data Structures & Algorithms in Java|edition=4th|first1=Michael T.|last1=Goodrich|first2=Roberto|last2=Tamassia|publisher=John Wiley & Sons|year=2006|isbn=978-0-471-73884-8}} See "Quadratic probing", p. 382, and exercise C–9.9, p. 415.</ref>

Some [[Checksum|checksum]] methods are based on the mathematics of prime numbers. For instance the checksums used in [[International Standard Book Number]]s 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>{{cite book|title=Identification Numbers and Check Digit Schemes|volume=18|series=Classroom Resource Materials|first=Joseph|last=Kirtland|publisher=Mathematical Association of America|year=2001|isbn=978-0-88385-720-5|pages=43–44|url=https://books.google.com/books?id=Z8eka35WUb8C&pg=PA43}}</ref> Another checksum method, [[Adler-32]], uses arithmetic modulo 65521, the largest prime number less than <math>2^{16}</math>.<ref>{{cite book|url=http://www.rfc-editor.org/rfc/rfc1950.txt|title=ZLIB Compressed Data Format Specification version 3.3|last=Deutsch|first=P.|publisher=Network Working Group|year=1996|series=Request for Comments|volume=1950}}</ref>
Prime numbers are also used in [[Pseudorandom number generator|pseudorandom number generator]]s including [[Linear congruential generator|linear congruential generator]]s<ref>{{cite book|title=The Art of Computer Programming, Vol. 2: Seminumerical algorithms|edition=3rd|first=Donald E.|last=Knuth|publisher=Addison-Wesley|year=1998|contribution=3.2.1 The linear congruential model|pages=10–26|isbn=978-0-201-89684-8|title-link=The Art of Computer Programming}}</ref> and the [[Mersenne Twister]].<ref>{{cite journal | last1 = Matsumoto | first1 = Makoto | last2 = Nishimura | first2 = Takuji | doi = 10.1145/272991.272995 | issue = 1 | journal = ACM Transactions on Modeling and Computer Simulation | pages = 3–30 | title = Mersenne Twister: A 623-dimensionally equidistributed uniform pseudo-random number generator | volume = 8 | year = 1998| citeseerx = 10.1.1.215.1141 | s2cid = 3332028 }}</ref>

== Other applications ==
Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including [[Abstract algebra|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-in-line problem|no three are in a line]], or so that every triangle formed by three of the points [[Heilbronn triangle problem|has large area]].<ref>{{cite journal | last = Roth | first = K.F. |  doi = 10.1112/jlms/s1-26.3.198 | journal = Journal of the London Mathematical Society | mr = 0041889 | pages = 198–204 | series = Second Series | title = On a problem of Heilbronn | volume = 26 | issue = 3 | year = 1951}}</ref> Another example is [[Eisenstein's criterion]], a test for whether a [[Irreducible polynomial|polynomial is irreducible]] based on divisibility of its coefficients by a prime number and its square.<ref>{{cite journal | first = David A. | last = Cox | author-link = David A. Cox | title = Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first | journal = [[American Mathematical Monthly]] | volume = 118 | issue = 1 | year = 2011 | pages = 3–31 | doi=10.4169/amer.math.monthly.118.01.003 | url = https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Cox-2012.pdf| citeseerx = 10.1.1.398.3440 | s2cid = 15978494 }}</ref>

[[File:Sum of knots3.svg|thumb|The connected sum of two prime knots]]
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|finite field]] with a prime number of elements, whence the name.<ref>{{cite book | last1=Lang | first1=Serge |  title=Algebra | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin; New York | series=Graduate Texts in Mathematics | isbn=978-0-387-95385-4 | mr=1878556 | year=2002 | volume=211| doi=10.1007/978-1-4613-0041-0 }}, 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|knot theory]], a [[Prime knot|prime knot]] is a knot that is indecomposable in the sense that it cannot be written as the [[Connected sum|connected sum]] of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.<ref>{{cite journal|last=Schubert|first= Horst|title=Die eindeutige Zerlegbarkeit eines Knotens in Primknoten|journal=S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl.|year=1949|volume=1949|issue=3|pages=57–104|mr=0031733}}</ref> The [[Prime decomposition (3-manifold)|prime decomposition of 3-manifolds]] is another example of this type.<ref>{{cite journal | last = Milnor | first = J. |  doi = 10.2307/2372800 | journal = American Journal of Mathematics | mr = 0142125 | pages = 1–7 | title = A unique decomposition theorem for 3-manifolds | volume = 84 | issue = 1 | year = 1962| jstor = 2372800 }}</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 [[Biology:Evolutionary biology|evolutionary biology]] to explain the life cycles of [[Biology:Cicada|cicada]]s.

===Constructible polygons and polygon partitions===
[[File:Pentagon construct.gif|thumb|Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a Fermat prime.|alt=Construction of a regular pentagon using straightedge and compass]]
Fermat primes are primes of the form
:<math>F_k = 2^{2^k}+1,</math>
with <math>k</math> a nonnegative integer.<ref>{{harvtxt|Boklan|Conway|2017}} also include <math>2^0+1=2</math>, which is not of this form.</ref> They are named after [[Biography:Pierre de Fermat|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">{{cite book | last1 = Křížek | first1 = Michal | last2 = Luca | first2 = Florian | last3 = Somer | first3 = Lawrence | doi = 10.1007/978-0-387-21850-2 | isbn =978-0-387-95332-8 | location = New York | mr = 1866957 | pages = 1–2 | publisher = Springer-Verlag | series = CMS Books in Mathematics | title = 17 Lectures on Fermat Numbers: From Number Theory to Geometry | url = https://books.google.com/books?id=hgfSBwAAQBAJ&pg=PA1 | volume = 9 | year = 2001}}</ref> but <math>F_5</math> is composite and so are all other Fermat numbers that have been verified as of 2017.<ref>{{cite journal | last1 = Boklan | first1 = Kent D. | last2 = Conway | first2 = John H. |  arxiv = 1605.01371 | date = January 2017 | doi = 10.1007/s00283-016-9644-3 | issue = 1 | journal = [[The Mathematical Intelligencer]] | pages = 3–5 | title = Expect at most one billionth of a new Ferma''t'' prime! | volume = 39 | s2cid = 119165671 }}</ref> A [[Regular polygon|regular <math>n</math>-gon]] is [[Constructible polygon|constructible using straightedge and compass]] if and only if the odd prime factors of <math>n</math> (if any) are distinct Fermat primes.<ref name="kls"/> Likewise, a regular <math>n</math>-gon may be constructed using straightedge, compass, and an [[Angle trisection|angle trisector]] if and only if the prime factors of [[Regular polygon|<math>n</math>]] are any number of copies of 2 or 3 together with a (possibly empty) set of distinct [[Pierpont prime]]s, primes of the form <math>2^a3^b+1</math>.<ref>{{cite journal | last = Gleason | first = Andrew M. |  doi = 10.2307/2323624 | issue = 3 | journal = [[American Mathematical Monthly]] | mr = 935432 | pages = 185–194 | title = Angle trisection, the heptagon, and the triskaidecagon | volume = 95 | year = 1988| jstor = 2323624 }}</ref>

It is possible to partition any convex polygon into <math>n</math> smaller convex polygons of equal area and equal perimeter, when <math>n</math> is a [[Prime power|power of a prime number]], but this is not known for other values of <math>n</math>.<ref>{{cite journal | last = Ziegler | first = Günter M. |  issue = 95 | journal = European Mathematical Society Newsletter | mr = 3330472 | pages = 25–31 | title = Cannons at sparrows | year = 2015}}</ref>

===Quantum mechanics===
Beginning with the work of [[Biography:Hugh Lowell Montgomery|Hugh Montgomery]] and [[Biography:Freeman Dyson|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 [[Physics:Quantum system|quantum system]]s.<ref>{{cite web|first=Ivars|last=Peterson|work=MAA Online |url=http://www.maa.org/mathland/mathtrek_6_28_99.html |title=The Return of Zeta |date=June 28, 1999 |access-date=2008-03-14 |url-status=dead |archive-url=https://web.archive.org/web/20071020141624/http://maa.org/mathland/mathtrek_6_28_99.html |archive-date=October 20, 2007 }}</ref><ref>{{Cite journal|last=Hayes|first=Brian|author-link=Brian Hayes (scientist)|date=2003|title=Computing science: The spectrum of Riemannium|jstor=27858239|journal=American Scientist|volume=91|issue=4|pages=296–300|doi=10.1511/2003.26.3349|s2cid=16785858 }}</ref> Prime numbers are also significant in [[Quantum information science|quantum information science]], thanks to mathematical structures such as [[Mutually unbiased bases|mutually unbiased bases]] and [[SIC-POVM|symmetric informationally complete positive-operator-valued measures]].<ref>{{Cite book|title=Geometry of quantum states : an introduction to quantum entanglement|last1=Bengtsson|first1=Ingemar|last2=Życzkowski|first2=Karol|publisher={{wipe|Cambridge University Press}}|year=2017|isbn=978-1-107-02625-4|edition=Second|location=Cambridge|pages=313–354|oclc=967938939|author-link2=Karol Życzkowski}}</ref><ref>{{Cite journal|last=Zhu|first=Huangjun|title=SIC POVMs and Clifford groups in prime dimensions|url=http://stacks.iop.org/1751-8121/43/i=30/a=305305?key=crossref.45cb006b9f3c7e510461594ea8dfa7f7|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=30|pages=305305|arxiv=1003.3591|doi=10.1088/1751-8113/43/30/305305|bibcode=2010JPhA...43D5305Z|year=2010|s2cid=118363843}}</ref>

===Biology===
The evolutionary strategy used by [[Biology:Cicada|cicada]]s of the genus ''[[Biology:Magicicada|Magicicada]]'' makes use of prime numbers.<ref>{{cite journal |last1=Goles |first1=E. |last2=Schulz |first2=O. |first3=M. |last3=Markus |year=2001 |title=Prime number selection of cycles in a predator-prey model |journal=Complexity |volume=6 |issue=4 |pages=33–38 |doi=10.1002/cplx.1040 |bibcode=2001Cmplx...6d..33G }}</ref> These insects spend most of their lives as [[Biology:Larva|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>{{cite journal |first1=Paulo R.A.|last1=Campos |first2=Viviane M.|last2=de Oliveira |first3=Ronaldo|last3=Giro |first4=Douglas S.|last4=Galvão | journal = [[Physics:Physical Review Letters|Physical Review Letters]] | title = Emergence of prime numbers as the result of evolutionary strategy | volume = 93 | doi = 10.1103/PhysRevLett.93.098107 |pmid=15447148 | issue = 9 | year = 2004 | page = 098107 | bibcode=2004PhRvL..93i8107C|arxiv = q-bio/0406017 |s2cid=88332 }}</ref><ref>{{cite news |newspaper=[[Finance:The Economist|The Economist]]| url=http://economist.com/PrinterFriendly.cfm?Story_ID=2647052 |title=Invasion of the Brood |date=May 6, 2004|access-date=2006-11-26 }}</ref>
In contrast, the multi-year periods between flowering in [[Biology:Bamboo|bamboo]] plants are hypothesized to be [[Smooth number|smooth number]]s, having only small prime numbers in their factorizations.<ref>{{cite magazine|last1=Zimmer|first1=Carl|date=May 15, 2015|title=Bamboo Mathematicians|url=https://www.nationalgeographic.com/science/article/bamboo-mathematicians|department=Phenomena: The Loom|magazine=National Geographic|access-date=February 22, 2018}}</ref>

===Arts and literature===
Prime numbers have influenced many artists and writers.
The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as ''La Nativité du Seigneur'' (1935) and ''Quatre études de rythme'' (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in 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>{{cite book | editor1-last=Hill | editor1-first=Peter Jensen | title=The Messiaen companion | publisher=Amadeus Press | location=Portland, OR | isbn=978-0-931340-95-6 | year=1995 | url = https://books.google.com/books?id=7ag3ymWqvfgC&pg=PT225 | at = Ex. 13.2 ''Messe de la Pentecôte'' 1 'Entrée'}}</ref>

In his science fiction novel ''Contact'', scientist [[Biography:Carl Sagan|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>{{cite book | last = Pomerance | first = Carl |  editor1-last = Hayes | editor1-first = David F. | editor2-last = Ross | editor2-first = Peter | contribution = Prime Numbers and the Search for Extraterrestrial Intelligence | contribution-url = https://gauss.dartmouth.edu/~carlp/PDF/extraterrestrial.pdf | isbn = 978-0-88385-548-5 | location = Washington, DC | mr = 2085842 | pages = 3–6 | publisher = Mathematical Association of America | series = MAA Spectrum | title = Mathematical Adventures for Students and Amateurs | year = 2004}}</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 [[Medicine:Asperger syndrome|Asperger syndrome]].<ref>{{cite news|url=https://www.theguardian.com/science/punctuated-equilibrium/2010/sep/16/curious-incident-dog-night-time|title=The Curious Incident of the Dog in the Night-Time|author=GrrlScientist|date=September 16, 2010|newspaper={{wipe|The Guardian}}|access-date=February 22, 2010|department=Science}}</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>{{cite news|newspaper={{wipe|The New York Times}}|title=Counting on Each Other|first=Liesl|last=Schillinger|date=April 9, 2010|url=https://www.nytimes.com/2010/04/11/books/review/Schillinger-t.html|department=Sunday Book Review}}</ref>

==Notes==
{{notelist}}

==References==
{{reflist}}

==External links==
* {{SpringerEOM|title=Prime number|id=p/p074530|mode=cs1}}
* Caldwell, Chris, The [[Prime Pages]] at [http://primes.utm.edu/ primes.utm.edu].
* [http://plus.maths.org/issue49/package/index.html 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===
* [http://www.javascripter.net/math/calculators/primefactorscalculator.htm Prime factors calculator] can factorize any positive integer up to 20 digits.
* [http://www.alpertron.com.ar/ECM.HTM Fast Online primality test with factorization] makes use of the Elliptic Curve Method (up to thousand-digits numbers, requires Java).
* [http://www.bigprimes.net/ Huge database of prime numbers]
* [http://www.primos.mat.br/indexen.html Prime Numbers up to 1 trillion] {{Webarchive|url=https://web.archive.org/web/20210227001026/http://www.primos.mat.br/indexen.html |date=2021-02-27 }}

{{Number theory}}
{{Divisor classes}}
{{Prime number classes}}
{{Classes of natural numbers}}

[[Category:Prime numbers| ]]
[[Category:Articles containing proofs]]
[[Category:Integer sequences]]

{{Sourceattribution|Prime number}}