Prove that for every different parity numbers $a,b \in \Bbb N$ there exist $c \in \Bbb Z$ such that numbers $a+c, b+c, ab+c$ are perfect squares.
I tried to find separate solutions, $a=2, b=7, c=2$ also $a=6, b=13, c=3$ and etc., then all $a+c, b+c, ab+c$ are perfect squares. But have no idea how to start proof in general.
Have any ideas how to start proof? Thanks.