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?

Do you mean for these to be positive integers? Otherwise $a=b=c=0$ is a trivial solution... — Clayton, Dec 10 '12 at 03:58
Sorry, I meant nonzero solutions. All three integers should be nonzero — Steven-Owen, Dec 10 '12 at 03:59

score 7 · Accepted Answer · answered Dec 10 '12 at 04:00

7

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

answered Dec 10 '12 at 04:00

Gerry Myerson

185,413

score 1 · Answer 2 · answered Dec 10 '12 at 06:26

1

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

gives explanation about this euler brick.

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

answered Dec 10 '12 at 06:26

Harish Kayarohanam

1,980
2
16
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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.

2 Answers2

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