Disprove the statement “There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square”.
$\displaystyle n^2+(n+1)^2 = y^2$
Thus,
$\displaystyle 2n^2+2n+(1-y^2)=0$
We require the distriminant to be a square.
$\displaystyle 4-2(1-y^2) = x^2$
$\displaystyle 4-2+2y^2 = x^2$
$\displaystyle x^2-2y^2 = 2$.
This is a Pellain equation-look-a-like.
If it has 1 solution it has infinitely many.
Indeed it does.