Proof about difference of squares

[cragar]

Homework Statement

Show that an integer n can be represented as a difference of 2 squares if it is either
odd or divisible by 4, otherwise not. The representation is unique if and only if n is a prime number.

The Attempt at a Solution

let x and y be integers so then we have $x^2-y^2=n=(x-y)(x+y)$
we would look at the case where x and y are even, then we could factor a 2 out of x-y and x+y so it would be divisible by 4. if x or y was odd then n would be odd. and if x and y were both odd we could factor a 2 out of x-y and x+y. But I am not sure how to prove the part
where n is prime, and that would imply x and y are unique. It seems that if n is prime
then x-y or x+y has to be 1 or else n would be composite.

 

[haruspex]

the case where x and y are even, then we could factor a 2 out of x-y and x+y so it would be divisible by 4.

True, but it isn't x and y both being even that really characterises this case.

if x or y was odd then n would be odd.

No, think that through again.