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?

Well, it's obvious isn't it?
How could two consecutive numbers of that size be squares? Write $a = 2^{500}+15$  and $2^{500}+16=a+1$. Imagine $a$ were a square. Then its square root would be a pretty huge number. If you so much as add $1$ to that number, the square of the result will be massively larger than $a$, there'll be no way to get something that squares to $a+1$.
That's the gist of the other answers. The first step in many proofs is to think about why the answer is obvious.
