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An '''elementary proof''' or '''elementary technique''' in mathematics is a [[proof]] that uses only [[real numbers]] or [[real analysis]] rather than the use of [[complex analysis]].<ref>http://mathworld.wolfram.com/ElementaryProof.html</ref>  

An '''elementary proof''' or '''elementary technique''' in mathematics is a [[proof]] that uses only [[real numbers]] or [[real analysis]] rather than the use of [[complex analysis]]<ref>http://mathworld.wolfram.com/ElementaryProof.html</ref> or reliance on less rigorous axioms, such as the [[Axiom of Choice]].  An elementary proof typically cannot be improved by expressing it in simpler form.

The [[Prime Number Theorem]] has long been proven using complex analysis ([[Riemann Zeta function]]), but in 1949 and 1950 an elementary proof by [[Paul Erdos]] and [[Atle Selberg]] earned Selberg the highest prize in math, the [[Fields Medal]]. In contrast, [[Andrew Wiles]]' proof of [[Fermat's Last Theorem]] did not use elementary techniques.<ref name="Occam">http://www.occampress.com/fermat.pdf Page 5</ref>

The [[Prime Number Theorem]] has long been proven using complex analysis ([[Riemann Zeta function]]), but in 1949 and 1950 an elementary proof by [[Paul Erdos]] and [[Atle Selberg]] earned Selberg the highest prize in math, the [[Fields Medal]]. In contrast, [[Andrew Wiles]]' proof of [[Fermat's Last Theorem]] is not an elementary proof.<ref name="Occam">http://www.occampress.com/fermat.pdf Page 5</ref>

===References===

== Significance of Elementary Proofs ==

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[[Category:Mathematics]]

[[Category:Mathematics]]