# Show that there exist no positive integers ...

• August 23rd 2011, 04:33 AM
alexmahone
Show that there exist no positive integers ...
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.
• August 23rd 2011, 10:35 AM
melese
Re: Show that there exist no positive integers ...
There's a theorem by Fermat: The equation $x^4-y^4=z^2$ is not solvable in nonzero integers.
• August 23rd 2011, 04:49 PM
alexmahone
Re: Show that there exist no positive integers ...
Quote:

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.