# Math Help - Diopantine equation #2: y^2+y+1=x^2.

1. ## Diopantine equation #2: y^2+y+1=x^2.

Show that y^2+y+1=x^2 has no positive integer solution.

2. Originally Posted by dila
Show that y^2+y+1=x^2 has no positive integer solution.
$x^2=y(y+1)+1$

Clearly, $x^2$ is odd.

$\implies x$ is odd.

$\implies x^2-1$ is a multiple of 8.

$\implies y(y+1)$ is a multiple of 8.

$\implies y\ or\ y+1$ is a multiple of 8.

Can you proceed? (because I can't!)

3. New solution:

$y^2, which implies that there is a perfect square between two consecutive perfect squares, which is impossible.