Positive integers that are differences of squares

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.

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.

Re: Positive integers that are differences of squares

Quote:

Originally Posted by

**Zalren** 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.

although the question is far from solved i could help you 'clean up'. :)

Re: Positive integers that are differences of squares