WLOG, let .

From considerations modulo 2, a and b must both be even, or both be odd.

We can rewrite k = (a + b)(a - b)

Both factors are even.

Thus 4 divides k.

Sorry, forgot about the "if" and only did the "only if." Hmm.

Okay, for the other part, let

It is not hard to show that n must have two factors with an even difference. Just find two factors of k, as in and take .

So we can take the midpoint of the two even factors, which would allow us to write .