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Upsetting Pitagoras

Age 14 to 18
Challenge Level

Take a cue from the title!


Can you in any way use $a$, $b$ and $c$ where you know $a^2 + b^2 = c^2$?


No knowledge is needed here, only mathematical reasoning. It is an important point to make that finding a method for producing solutions does not prove that there is no other method and no smaller solution generated another way. However, once you have found what you believe to be the smallest solution, it is possible to check numerically that there are in fact no smaller solutions.