Discrete Mathematics

Question

Prove that if n is a perfect square, then n + 2 is not a perfect square.

Answer 1

All perfect squares are 0 or 1 mod 4. This is easy to show. If an integer z is even, it may be written as 2n, n an integer. Then, (2n)^2 = 4n^2. 4 |4n^2, so the perfect square mod 4 = 0

If an integer z is odd, it may be written as 2n+1. Then, (2n+1)^2 = 4n^2 + 4n + 1 = 4(n^2 _+ n) + 1, so the perfect square mod n = 1.

If x mod 4 = 0 or 1, then x+2 mod 4 = 2 or 3. Thus, x+2 mod 4 is not equal to 0 or 1, so it can't be a perfect square.