Editing Number theory (section)

== Applications ==
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 other than the use of prime numbered gear teeth to distribute wear evenly.<ref>{{cite book |last1=Bryant |first1=John |title=How Round is Your Circle?: Where Engineering and Mathematics Meet |title-link=How Round Is Your Circle |last2=Sangwin |first2=Christopher J. |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 [[United Kingdom|British]] mathematician [[G. H. Hardy]] prided themselves on doing work that had absolutely no military significance.<ref>{{cite book |last1=Hardy |first1=Godfrey Harold |author1-link=G. H. Hardy |title=A Mathematician's Apology |title-link=A Mathematician's Apology |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}}</ref> The number-theorist [[Leonard Dickson]] (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory.<ref>''The Unreasonable Effectiveness of Number Theory'', Stefan Andrus Burr, George E. Andrews, American Mathematical Soc., 1992, {{isbn|978-0-8218-5501-0}}</ref> 

This vision of the purity of number theory was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of [[public-key cryptography]] algorithms.<ref>{{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> Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors.<ref>{{Cite book |url=https://www.taylorfrancis.com/books/9781351664110 |title=An Introduction to Number Theory with Cryptography |date=2018 |publisher=Chapman and Hall/CRC |isbn=978-1-351-66411-0 |edition=2nd |doi=10.1201/9781351664110 |access-date=2023-02-22 |archive-url=https://web.archive.org/web/20230301144259/https://www.taylorfrancis.com/books/mono/10.1201/9781351664110/introduction-number-theory-cryptography-james-kraft-lawrence-washington |archive-date=2023-03-01 |url-status=live}}</ref> These applications have led to significant study of [[Algorithm|algorithms]] for computing with prime numbers, and in particular of [[Primality test|primality testing]], methods for determining whether a given number is prime. Prime numbers are also used in computing for [[Checksum|checksums]], [[Hash table|hash tables]], and [[Pseudorandom number generator|pseudorandom number generators]].

In 1974, [[Donald Knuth]] said "virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".<ref>Computer science and its relation to mathematics" DE Knuth – The American Mathematical Monthly, 1974</ref>
Elementary number theory is taught in [[discrete mathematics]] courses for [[computer scientist]]s. It also has applications to the continuous in [[numerical analysis]].<ref>"Applications of number theory to numerical analysis", Lo-keng Hua, Luogeng Hua, Yuan Wang, Springer-Verlag, 1981, {{isbn|978-3-540-10382-0}}</ref>

Number theory has now several modern applications spanning diverse areas such as:
* [[Computer science]]: The [[fast Fourier transform]] (FFT) algorithm, which is used to efficiently compute the discrete Fourier transform, has important applications in signal processing and data analysis.<ref>{{cite book | last=Krishna | first=Hari | title=Digital Signal Processing Algorithms | publisher=Routledge | date=2017 | location=London | isbn=978-1-351-45497-1}}</ref>
* [[Physics]]: The [[Riemann hypothesis]] has connections to the distribution of prime numbers and has been studied for its potential implications in physics.<ref>{{cite journal |title=Physics of the Riemann Hypothesis |journal=Reviews of Modern Physics |volume=83 |issue=2 |pages=307–330 |first1=Daniel |last1=Schumayer |first2=David A. W. |last2=Hutchinson |year=2011 |arxiv=1101.3116 |doi=10.1103/RevModPhys.83.307 |bibcode=2011RvMP...83..307S |s2cid=119290777}}</ref>
* [[Error correction code]]s: The theory of finite fields and algebraic geometry have been used to construct efficient error-correcting codes.<ref>{{Cite book |last=Baylis |first=John |url=https://www.taylorfrancis.com/books/9781351449847 |title=Error-Correcting Codes: A Mathematical Introduction |date=2018 |publisher=Routledge |isbn=978-0-203-75667-6 |doi=10.1201/9780203756676 |access-date=2023-02-22}}</ref>
* Communications: The design of cellular telephone networks requires knowledge of the theory of [[modular form]]s, which is a part of analytic number theory.<ref>{{Citation |last=Livné |first=R. |title=Communication Networks and Hilbert Modular Forms |date=2001 |url=http://link.springer.com/10.1007/978-94-010-1011-5_13 |work=Applications of Algebraic Geometry to Coding Theory, Physics and Computation |pages=255–270 |editor-last=Ciliberto |editor-first=Ciro |place=Dordrecht |publisher=Springer |doi=10.1007/978-94-010-1011-5_13 |isbn=978-1-4020-0005-8 |access-date=2023-02-22 |editor2-last=Hirzebruch |editor2-first=Friedrich |editor3-last=Miranda |editor3-first=Rick |editor4-last=Teicher |editor4-first=Mina}}</ref>
* Study of musical scales: the concept of "[[equal temperament]]", which is the basis for most modern Western music, involves dividing the [[octave]] into 12 equal parts.<ref>{{Cite journal |last1=Cartwright |first1=Julyan H. E. |last2=Gonzalez |first2=Diego L. |last3=Piro |first3=Oreste |last4=Stanzial |first4=Domenico |date=2002-03-01 |title=Aesthetics, Dynamics, and Musical Scales: A Golden Connection |url=http://dx.doi.org/10.1076/jnmr.31.1.51.8099 |journal=Journal of New Music Research |volume=31 |issue=1 |pages=51–58 |doi=10.1076/jnmr.31.1.51.8099 |hdl=10261/18003 |s2cid=12232457 |issn=0929-8215 |hdl-access=free}}</ref> This has been studied using number theory and in particular the properties of the 12th root of 2.