Editing Mathematics (section)

=== Number theory ===
{{Main|Number theory}}
[[File:Spirale Ulam 150.jpg|thumb|This is the [[Ulam spiral]], which illustrates the distribution of [[prime numbers]]. The dark diagonal lines in the spiral hint at the hypothesized approximate [[Independence (probability theory)|independence]] between being prime and being a value of a quadratic polynomial, a conjecture now known as [[Ulam spiral#Hardy and Littlewood's Conjecture F|Hardy and Littlewood's Conjecture F]].]]
Number theory began with the manipulation of [[number]]s, that is, [[natural number]]s <math>(\mathbb{N}),</math> and later expanded to [[integer]]s <math>(\Z)</math> and [[rational number]]s <math>(\Q).</math> Number theory was once called arithmetic, but nowadays this term is mostly used for [[numerical calculation]]s.<ref>{{cite book |last=LeVeque |first=William J. |author-link=William J. LeVeque |year=1977 |chapter=Introduction |title=Fundamentals of Number Theory |pages=1–30 |publisher=[[Addison-Wesley Publishing Company]] |isbn=0-201-04287-8 |lccn=76055645 |oclc=3519779 |s2cid=118560854}}</ref> Number theory dates back to ancient [[Babylonian mathematics|Babylon]] and probably [[ancient China|China]]. Two prominent early number theorists were [[Euclid]] of ancient Greece and [[Diophantus]] of Alexandria.<ref>{{cite book |last=Goldman |first=Jay R. |year=1998 |chapter=The Founding Fathers |title=The Queen of Mathematics: A Historically Motivated Guide to Number Theory |pages=2–3 |publisher=A K Peters |publication-place=Wellesley, MA |doi=10.1201/9781439864623 |isbn=1-56881-006-7 |lccn=94020017 |oclc=30437959 |s2cid=118934517}}</ref> The modern study of number theory in its abstract form is largely attributed to [[Pierre de Fermat]] and [[Leonhard Euler]]. The field came to full fruition with the contributions of [[Adrien-Marie Legendre]] and [[Carl Friedrich Gauss]].<ref>{{cite book |last=Weil |first=André |author-link=André Weil |year=1983 |title=Number Theory: An Approach Through History From Hammurapi to Legendre |publisher=Birkhäuser Boston |pages=2–3 |doi=10.1007/978-0-8176-4571-7 |isbn=0-8176-3141-0 |lccn=83011857 |oclc=9576587 |s2cid=117789303}}</ref>

Many easily stated number problems have solutions that require sophisticated methods, often from across mathematics. A prominent example is [[Fermat's Last Theorem]]. This conjecture was stated in 1637 by Pierre de Fermat, but it [[Wiles's proof of Fermat's Last Theorem|was proved]] only in 1994 by [[Andrew Wiles]], who used tools including [[scheme theory]] from [[algebraic geometry]], [[category theory]], and [[homological algebra]].<ref>{{cite journal |last=Kleiner |first=Israel |author-link=Israel Kleiner (mathematician) |date=March 2000 |title=From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem |journal=[[Elemente der Mathematik]] |volume=55 |issue=1 |pages=19–37 |doi=10.1007/PL00000079 |doi-access=free |issn=0013-6018 |eissn=1420-8962 |lccn=66083524 |oclc=1567783 |s2cid=53319514}}</ref> Another example is [[Goldbach's conjecture]], which asserts that every even integer greater than 2 is the sum of two [[prime number]]s. Stated in 1742 by [[Christian Goldbach]], it remains unproven despite considerable effort.<ref>{{cite book |last=Wang |first=Yuan |year=2002 |title=The Goldbach Conjecture | pages=1–18 |edition=2nd |series=Series in Pure Mathematics |volume=4 |publisher=[[World Scientific]] |doi=10.1142/5096 |isbn=981-238-159-7 |lccn=2003268597 |oclc=51533750 |s2cid=14555830}}</ref>

Number theory includes several subareas, including [[analytic number theory]], [[algebraic number theory]], [[geometry of numbers]] (method oriented), [[diophantine equation]]s, and [[transcendence theory]] (problem oriented).<ref name=MSC />