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How would you go about finding three nonzero integers $a,\, b,\,$ and $c$ such that $\sqrt{a^2+b^2}$, $\sqrt{a^2+c^2}$, $\sqrt{c^2+b^2}$, and $\sqrt{a^2+b^2+c^2}$ are all integers? Does anyone know if this is not solvable, and if so, is there an elementary proof of it?

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Do you mean for these to be positive integers? Otherwise $a=b=c=0$ is a trivial solution... – Clayton Dec 10 '12 at 3:58
Sorry, I meant nonzero solutions. All three integers should be nonzero – Steven-Owen Dec 10 '12 at 3:59
up vote 7 down vote accepted

This is the "integer cuboid" or "Euler brick" problem. Currently wide open.

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