Editing Prime number (section)

== Other applications ==
Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including [[abstract algebra]] and elementary geometry. For example, it is possible to place prime numbers of points in a two-dimensional grid so that [[no-three-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=Klaus F. |author-link=Klaus Roth |year=1951 |title=On a problem of Heilbronn |journal=[[Journal of the London Mathematical Society]] |series=Second Series |volume=26 |issue=3 |pages=198–204 |doi=10.1112/jlms/s1-26.3.198 |mr=0041889}}</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 | access-date = 2018-01-25 | archive-date = 2023-03-26 | archive-url = https://web.archive.org/web/20230326032030/https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Cox-2012.pdf | url-status = dead }}</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]] with a prime number of elements, whence the name.<ref>{{cite book |last1=Lang |first1=Serge |author1-link=Serge Lang |title=Algebra |publisher=[[Springer-Verlag]] |year=2002 |isbn=978-0-387-95385-4 |series=Graduate Texts in Mathematics |volume=211 |location=Berlin, Germany; New York |doi=10.1007/978-1-4613-0041-0 |mr=1878556}} Section II.1, p. 90.</ref> Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in [[knot theory]], a [[prime knot]] is a [[knot (mathematics)|knot]] that is indecomposable in the sense that it cannot be written as the [[connected sum]] of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.<ref>{{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. | author-link = John Milnor | 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 mechanic]]s, and have been used metaphorically in the arts and literature. They have also been used in [[evolutionary biology]] to explain the life cycles of [[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 prime]]s are primes of the form
: <math>F_k = 2^{2^k}+1,</math>
with {{tmath|k}} a [[nonnegative integer]].<ref>{{harvtxt|Boklan|Conway|2017}} also include {{tmath|1= 2^0+1=2 }}, which is not of this form.</ref> They are named after [[Pierre de Fermat]], who conjectured that all such numbers are prime. The first five of these numbers – 3, 5, 17, 257, and 65,537 – are prime,<ref name="kls">{{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. | author2-link = John Horton Conway | 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 {{tmath|n}}-gon]] is [[constructible polygon|constructible using straightedge and compass]] if and only if the odd prime factors of {{tmath|n}} (if any) are distinct Fermat primes.<ref name="kls"/> Likewise, a regular {{tmath|n}}-gon may be constructed using straightedge, compass, and an [[Angle trisection|angle trisector]] if and only if the prime factors of [[regular polygon|{{tmath|n}}]] are any number of copies of 2 or 3 together with a (possibly empty) set of distinct [[Pierpont prime]]s, primes of the form {{tmath|2^a3^b+1}}.<ref>{{cite journal | last = Gleason | first = Andrew M. | author-link = Andrew M. Gleason | 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 {{tmath|n}} smaller convex polygons of equal area and equal perimeter, when {{tmath|n}} is a [[prime power|power of a prime number]], but this is not known for other values of {{tmath|n}}.<ref>{{cite journal | last = Ziegler | first = Günter M. | author-link = Günter M. Ziegler | 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 [[Hugh Lowell Montgomery|Hugh Montgomery]] and [[Freeman Dyson]] in the 1970s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of [[quantum system]]s.<ref>{{cite web|first=Ivars|last=Peterson|author-link=Ivars 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]], thanks to mathematical structures such as [[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 |title-link=Geometry of Quantum States |last1=Bengtsson |first1=Ingemar |last2=Życzkowski |first2=Karol |publisher=[[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 [[cicada]]s of the genus ''[[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 (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 [[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 |last1=Campos |first1=Paulo R. A. |last2=de Oliveira |first2=Viviane M. |last3=Giro |first3=Ronaldo |last4=Galvão |first4=Douglas S. |year=2004 |title=Emergence of prime numbers as the result of evolutionary strategy |journal=[[Physical Review Letters]] |volume=93 |issue=9 |page=098107 |arxiv=q-bio/0406017 |bibcode=2004PhRvL..93i8107C |doi=10.1103/PhysRevLett.93.098107 |pmid=15447148 |s2cid=88332}}</ref><ref>{{cite news |newspaper=[[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 [[bamboo]] plants are hypothesized to be [[smooth number]]s, having only small prime numbers in their factorizations.<ref>{{cite magazine|last1=Zimmer|first1=Carl|author-link=Carl Zimmer|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]]'' (1949–1950), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".<ref>{{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 (novel)|Contact]]'', scientist [[Carl Sagan]] suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer [[Frank Drake]] in 1975.<ref>{{cite book | last = Pomerance | first = Carl | author-link = Carl Pomerance | 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 [[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=[[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 (novel)|The Solitude of Prime Numbers]]'', in which they are portrayed as "outsiders" among integers.<ref>{{cite news|newspaper=[[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>