TITLE: How to prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?
QUESTION [20 upvotes]: How would I prove that either $2^{500} + 15$ or $2^{500} + 16$ isn't a perfect square?

REPLY [5 votes]: First write down exactly what it is that you're trying to prove is impossible.  In this case, that means writing down these two equations:
$$2^{500}+15=n^2$$
$$2^{500}+16=m^2$$
Next, play with the equations and see where they lead you.  If you subtract the first from the second, you get
$$1=m^2-n^2$$
or 
$$1=(m+n)(m-n)$$
Now think about what this tells you about $m$ and $n$.  Hint:  What are all the factors of $1$?