Hi,

If it is known that $\displaystyle (A^2+B^2)(C^2+D^2)$ is a perfect square

where A, B, C and D are integers, A and B are coprime, C and D are coprime and D > C >= B > A > 0

then $\displaystyle (A^2+B^2)(C^2+D^2)+4(A^2-B^2)CD$be a perfect square..........(1)can

and $\displaystyle \ (A^2+B^2)(C^2+D^2)-4(A^2-B^2)CD$be a perfect square..........(2)can

but can it be proven that (1) and (2)be perfect squares simultaneously (or examples to the contrary) ?can't

Examples

A = 1, B = 2, C = 2, D = 11

$\displaystyle (A^2+B^2)(C^2+D^2)=25^2$

$\displaystyle (A^2+B^2)(C^2+D^2)+4(A^2-B^2)CD=19^2$

$\displaystyle \ (A^2+B^2)(C^2+D^2)-4(A^2-B^2)CD$ is not a perfect square

A = 1, B = 2, C = 58, D = 209

$\displaystyle (A^2+B^2)(C^2+D^2)=485^2$

$\displaystyle (A^2+B^2)(C^2+D^2)+4(A^2-B^2)CD$ is not a perfect square

$\displaystyle \ (A^2+B^2)(C^2+D^2)-4(A^2-B^2)CD=617^2$

thank you