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?
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.
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number-theory
diophantine-equations
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0Sorry, I meant nonzero solutions. All three integers should be nonzero – 2012-12-10
2 Answers
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This is the "integer cuboid" or "Euler brick" problem. Currently wide open.
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http://mathworld.wolfram.com/EulerBrick.html
gives explanation about this euler brick.