How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square? How would I prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?

Alternative method (for fun):
Observe $2^4 = 16 \equiv 1$ mod $5$.
Then $2^{500} = (2^4)^{125} \equiv 1^{125} = 1$ mod $5$.
Thus, $2^{500} + 16 \equiv 1 + 1 = 2$ mod $5$.
But every square is either $0, 1$ or $4$ mod $5$.
Therefore, $2^{500} + 16$ cannot be a square. QED
