Proving that people still discover things about the oldests of proofs, this is from Geoffrey C. Berresford in 2008. [Claims abound of it being much older, so bonus kudos to whoever finds the oldest reference.]
It is almost stupidly easy (and hence is very smart).
Prove: If is a rational number then is a perfect square.
Suppose: is a rational number in lowest terms.
Doing a little algebra…
… we can say
Since is in lowest terms, there’s an integer such that and .
Since we also know . That is, is an integer, so is an integer and is a perfect square.
Taking the contrapositive, if is a not a perfect square then is not a rational number.
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