Prove that there exist no positive integers $m$ and $n$ for which $m^2+m+1=n^2$

Hint: If such $n$ existed, then it would have to be between $m$ and $m+1$ since $$m^2<m^2+m+1<m^2+2m+1=(m+1)^2.$$

Clayton's answer is certainly simpler that this, but here's an alternative.

Multiply both sides by $4$ and you get:

$$(2n)^2 = 4m^2+4m+4 = (2m+1)^2 + 3$$

So $$(2n-(2m+1))(2n+(2m+1)) = 3$$

This means that $3$ can be factored into two numbers that differ by $4m+2$. That means $m=0$ or $m=-1$.