I'm working on proving the following statement:
If n is a perfect square, then n+2 is not a perfect square.
I also need to state this in first order logic with arithmetic, but have no idea what that looks like.
The only start I have so far in terms of the proof is:
$n$ = $a^2$
$n+2$ = $b^2$
But I don't know how to proceed from here? I've seen solutions to this already but do not understand how they actually prove anything.
Best Answer
First, for $n = 1$:
$n = 1$: $n^2 = 1$, $n^2 + 2 = 3$, 3 is not a perfect square
For $n \ge 2$:
There are no perfect squares between $n^2$ and $(n + 1)^2$, exclusive. For $n \ge 2$, $n^2 < n^2 + 2 < (n + 1)^2$, so $n + 2$ is not a perfect square.