There don't exist $a, b$ positive integers such that $a^2 + b^2$ and $a^2 - b^2$ are perfect squares I need to prove that there don't exist $a, b$ positive integers such that $a^2 + b^2$ and $a^2 - b^2$ are perfect squares.
I suopose that $a^2 + b^2 = c^2$ and $a^2 - b^2 = d^2$ with $c, d$ positive integers, so 
$$(a^2 + b^2)(a^2 - b^2) = c^2d^2$$
Therefore
$$(a^2 + b^2)(a^2 - b^2) = (cd)^2$$
and 
$$a^4 - b^4 = (cd)^2$$
but this equation doesn't have solution in positive integers. 
Is that right?

If $a^4-b^4=(cd)^2$ then that yields $b^4+(cd)^2=a^4$ . @awllower proved here that there can never exist a right triangle with integer sides such that a leg and the hypotenuse can be perfect squares. 
