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?

The question is not about that big numbers...
Question is about a result that two consecutive numbers can not be squares simultaneously..
Suppose  $a=b^2\text { and }a+1=c^2\Rightarrow b^2+1=c^2\Rightarrow c^2-b^2=1\Rightarrow (c+b)(c-b)=1$
w.l.o.g. assume $c+b=1$ which implies 
$a+1=c^2=(1-b)^2=1+b^2-2b\Rightarrow a=b^2-2b$ but then we have $a=b^2$
you should now be able to see some contradiction...
So...
