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The question is simple: Find the longest possible hypotenuse in a right triangle with integer sides where the shortest side has length T. What I am asking is if there are any means to approach this besides testing pythagorean triples by brute force.

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Hint: If the given side is $T$, there will exist an $n$ such that $(n+1)^2-n^2=2n+1>T^2$.

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