Answer to Question #329607 in Discrete Mathematics for waji

Question

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

perfect square.

Answer 1

Let $n=k^2,$ where $k$ is some integer number.

Let's find the nearest to the $n$ number that is a perfect square and is greater than $n$.

Obviously, this number is $(k+1)^2=k^2+2k+1=n+2k+1$ and it is bigger than $n+2$ when $k\ge1$. So, $\sqrt{n+2}$ should lie between $k$ and $k+1$ and it is not an integer number.