Show that there exist no positive integers ...

**alexmahone** · August 23rd 2011, 04:33 AM

Show that there exist no positive integers $m$ and $n$ such that $m^2+n^2$ and $m^2-n^2$ are both perfect squares.

**melese** · August 23rd 2011, 10:35 AM

There's a theorem by Fermat: The equation $x^4-y^4=z^2$ is not solvable in nonzero integers.

**alexmahone** · August 23rd 2011, 04:49 PM

Originally Posted by melese

There's a theorem by Fermat: The equation $x^4-y^4=z^2$ is not solvable in nonzero integers.

Thanks!

So, $m^2+n^2=a^2$ and $m^2-n^2=b^2$ .

$(m^2+n^2)(m^2-n^2)=a^2b^2$

$m^4-n^4=(ab)^2$ , which has no solution.

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