As you pointed out, a2−b2=(a+b)(a−b). If a,b are both odd or both even, then a+b and a−b are both even and so their product is divisible by 4. If one of a,b is odd and the other even, then a+b and a−b are both odd and so their product is odd. Hence a2−b2 is always either odd or divisible by 4; it cannot be even and not divisible by 4.
This is perfect. Thank you.