First of all I want to say that this is not the namesake competition . We are not " you against me " but " infinity against infinity". This post is for all those who have either created or have perceived " the most beautiful solution ever " .I created this for the dreamers who may want to wander through the realms of mathematics in one alley having many shops . Please provide the link to that particular solution(s).

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TopNewestClaim: An irrational raised to the power of an irrational can be a rational.

Proof: Either \(\sqrt{2}^\sqrt{2} \) is irrational or rational.

If it is rational, we are done.

Otherwise, \({\sqrt{2}^\sqrt{2}}^\sqrt{2} =\sqrt{2}^2 = 2 \) is rational. – Agnishom Chattopadhyay · 1 year, 6 months ago

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\(\large (\sqrt{2}^{\sqrt{2}})^{\sqrt{2}} = \sqrt{2}^{(\sqrt{2}\times \sqrt{2})} = \sqrt{2}^{2} = 2,\)

since \(\large \sqrt{2}^{\sqrt{2}^{\sqrt{2}}} = 1.7608.....\) is irrational. – Brian Charlesworth · 1 year, 6 months ago

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– Raven Herd · 1 year, 6 months ago

Nice : )Log in to reply

Don Zagier's one sentence proof of Fermat's theorem on sums of two squares is quite beautiful. – Prasun Biswas · 1 year, 5 months ago

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