Editing Number theory (section)

{{Short description|Branch of mathematics}}
{{for|the book by André Weil|Number Theory: An Approach Through History from Hammurapi to Legendre{{!}}''Number Theory: An Approach Through History from Hammurapi to Legendre''}}
{{distinguish|Numerology}}
[[File:A 150x150 Ulam spiral of dots with varying widths (emphasis primes).svg|thumb|The distribution of [[prime number]]s, a central point of study in number theory, illustrated by an [[Ulam spiral]]. It shows the conditional [[Independence (probability theory)|independence]] between being prime and being a value of certain quadratic polynomials.]]
{{Math topics TOC}}
'''Number theory''' is a branch of [[pure mathematics]] devoted primarily to the study of the [[integer]]s and [[arithmetic function]]s. Number theorists study [[prime number]]s as well as the properties of [[mathematical object]]s constructed from integers (for example, [[rational number]]s), or defined as generalizations of the integers (for example, [[algebraic integer]]s). 

Integers can be considered either in themselves or as solutions to equations ([[Diophantine geometry]]). Questions in number theory can often be understood through the study of [[Complex analysis|analytical]] objects, such as the [[Riemann zeta function]], that encode properties of the integers, primes or other number-theoretic objects in some fashion ([[analytic number theory]]). One may also study [[real number]]s in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions ([[Diophantine approximation]]).

Number theory is one of the oldest branches of mathematics alongside geometry.  One quirk of number theory is that it deals with statements that are simple to understand but are very difficult to solve. Examples of this are [[Fermat's Last Theorem]], which was proved 358 years after the original formulation, and [[Goldbach's conjecture]], which remains unsolved since the 18th century. German mathematician [[Carl Friedrich Gauss]] (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."{{sfn|Long|1972|p=1}} It was regarded as the example of pure mathematics with no applications outside mathematics until the 1970s, when it became known that prime numbers would be used as the basis for the creation of [[public-key cryptography]] algorithms.