User:Alecscooper/Sandbox

Theorem
If $$n$$ is a perfect square other than $$0$$, then $$n+1$$ is not a perfect square.

Proof
If $$n$$ and $$n+1$$ are perfect squares, $$n\neq0$$, then $$\exists j,k\in \mathbb{N}: j^2=n, k^2=n+1$$.

Thus $$j^2-k^2=1$$.

So $$(j-k)(j+k)=1$$.

Since $$j$$ and $$k$$ are both integers and integers are closed under addition, $$j-k$$ and $$j+k$$ are integers.

It also follows that $$j-k$$ and $$j+k$$ are multiplicative inverses.

Since their product is positive, it follows that $$j-k$$ and $$j+k$$ have the same sign.

Since the natural numbers are closed under addition, $$(j+k)\in\mathbb{N}$$, so both $$j-k$$ and $$j+k$$ are positive.