(n^2)+(s^2) is not divisible by 4k-1, if and only if s=4m+1 and s is not divisible by 4k-1

Question

I know 4k-1 does not divide (n^2)+1, 4T+1, or (n^2)+[(n+1)^2], where T is a triangular number, though I still don't understand the proofs. How can I prove that, if s = 4m+1 (either prime or compound), (n^2)+(s^2) is not divisible by 4k-1, if and only if s is not divisible by 4k-1.