Mathematics Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! === 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|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/> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page