# Thread: Show that there exist no positive integers ...

1. ## Show that there exist no positive integers ...

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

2. ## Re: Show that there exist no positive integers ...

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

3. ## Re: Show that there exist no positive integers ...

Originally Posted by melese
There's a theorem by Fermat: The equation $\displaystyle x^4-y^4=z^2$ is not solvable in nonzero integers.
Thanks!

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

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

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