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

Then

.

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.