I haven't had any advance. I'd appreciate any ideas. Maybe if we assume that one of the numbers is a perfect square, so that
4(a^2) + r(b) = k^2, where k is a positive integer, then as a consequence,
4(b^2) + r(a) shouldn't be a perfect square. Or if we assume that both numbers are perfect squares, we would surely arrive at some contradiction. I have tried to do both ways, but I have got nowhere. All I have are some crumpled papers with nothing valuable in them. Maybe someone can show me how to prove this?