Editing Leonhard Euler (section)

===Number theory===
Euler's interest in number theory can be traced to the influence of [[Christian Goldbach]],{{sfn|Calinger|1996|p=130}} his friend in the St. Petersburg Academy.{{sfn|Gautschi|2008|p=6}} Much of Euler's early work on number theory was based on the work of [[Pierre de Fermat]]. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form <math display="inline">2^{2^n}+1</math> ([[Fermat numbers]]) are prime.{{sfn|Dunham|1999|p=7}}

Euler linked the nature of prime distribution with ideas in analysis. He proved that [[Proof that the sum of the reciprocals of the primes diverges|the sum of the reciprocals of the primes diverges]]. In doing so, he discovered the connection between the [[Riemann zeta function]] and prime numbers; this is known as the [[Proof of the Euler product formula for the Riemann zeta function|Euler product formula for the Riemann zeta function]].<ref name=patterson/>

Euler invented the [[totient function]] φ(''n''), the number of positive integers less than or equal to the integer ''n'' that are [[coprime]] to ''n''. Using properties of this function, he generalized [[Fermat's little theorem]] to what is now known as [[Euler's theorem]].<ref name=shiu/> He contributed significantly to the theory of [[perfect number]]s, which had fascinated mathematicians since [[Euclid]]. He proved that the relationship shown between even perfect numbers and [[Mersenne prime]]s (which he had earlier proved) was one-to-one, a result otherwise known as the [[Euclid–Euler theorem]].<ref name=stillwell/> Euler also conjectured the law of [[quadratic reciprocity]]. The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of [[Carl Friedrich Gauss]], particularly ''[[Disquisitiones Arithmeticae]]''.{{sfn|Dunham|1999|loc=Ch. 1, Ch. 4}} By 1772 Euler had proved that 2<sup>31</sup>&nbsp;−&nbsp;1 = [[2147483647|2,147,483,647]] is a Mersenne prime. It may have remained the [[largest known prime]] until 1867.<ref name=caldwell/>

Euler also contributed major developments to the theory of [[partitions of an integer]].<ref name=hopwil/>