Problem understanding a fact

Hello ! I have troubles understanding this statement :

Quote:

If one, in an attempt to factorize a composite number

, finds two squares

and

at random such as

, then taking the greatest common divisor of

with

yields a nontrivial factor of n with probability

.

I don't understand why there is 0.5 probability to find a nontrivial factor of , since if we call one factor of , then if we find a nontrivial factor, must divide , and so there are possibilities that this happens, whether if we find a trivial factor of , then , and there are infinitely many numbers and that satisfy this condition. So it should be very likely that we find a trivial factor of with a difference of squares, which is not true.

Can somebody explain to me where I went wrong in my reasoning ? Thanks !