All integers are of four different forms: 4x, 4x + 1, 4x + 2, and 4x + 3 for some integer x. Since squares of even numbers are even, 4x + 3 must be the square of an odd number, if it would be a square. All odd numbers are either of the form 4x + 1 or 4x + 3. Now show that and are both of the form 4x + 1, and thus 4x + 3 cannot be the square of any odd integer, and hence cannot be the square of any integer.