Show that y^2+y+1=x^2 has no positive integer solution.
$\displaystyle x^2=y(y+1)+1$
Clearly, $\displaystyle x^2$ is odd.
$\displaystyle \implies x$ is odd.
$\displaystyle \implies x^2-1$ is a multiple of 8.
$\displaystyle \implies y(y+1)$ is a multiple of 8.
$\displaystyle \implies y\ or\ y+1$ is a multiple of 8.
Can you proceed? (because I can't!)