I have a couple of questions 1 of which I more or less know what should be the proof, I just dont really know how to prove it.

1. My first question is how would I go about proving that m^2 = n^2

if and only if m = n or m = -n.

I know I have to prove it both ways, but am not quite sure how to construct it.

2. My next problem is proving that either 2x10^500 + 15 or 2x10^500 + 16

is not a perfect square.

Now I am fairly certain that perfect squares do not appear sequentially, thus proving that one of them is not square, but I am not sure how to exactly go about it.

The first question should probably be a proof that is fairly simple, these questions are both coming from chapters 1.6 and 1.7 of my book, so they shouldnt be too difficult.

help appreciated. Extra points to those who can help me understand a little bit more.