although the question is far from solved i could help you 'clean up'.Find all positive integers which can be written as the difference of squares.
This is what I have so far: Let be a positive integer such that . Also I believe I am safe in assuming that because the squares of the negatives are the same as the squares of the positives, and because if then is not a positive integer.
We have two cases.
Case 1) is even.
so either or is even.
If is even, then , for some . So
So if is even then is also even. So is even no matter if is even or not. Satisfying that is even.
the above said statement is correct but a concise and precise statement would be .
thus must be divisible by 4 if is even
Case 2) is odd.
This is where I am weak, I think.
for some . This implies .
if is odd.
So I think I can conclude that all the positive integers that can be written as the difference of squares is divisible by 4, if even, and greater than 3, if odd.
Any help correcting and/or cleaning up what I have would be appreciated.