Find three 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.

Question

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?

Answer 1

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

Answer 2

http://mathworld.wolfram.com/EulerBrick.html

gives explanation about this euler brick.

also http://en.wikipedia.org/wiki/Euler_brick#Properties