Case 1: x is even. Then we know where and the parity is different. It is safe to assume that is even (because the other way around everything is symettric except for but that not matter because we are squaring it, and thereby removing the negative sign). But . Which means, (remember is even). Since it means and . Thus, . For arbitrary relatively prime integers .

Case 2: x is odd. Then we know . Thus, . This forces to be even. And so, . Thus, .