Disprove the statement “There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square”.

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- Sep 16th 2007, 07:41 AMadnan0Disprove a statement
Disprove the statement “There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square”.

- Sep 16th 2007, 08:16 AMtopsquark
- Sep 16th 2007, 08:17 AMThePerfectHacker
- Sep 16th 2007, 08:19 AMThePerfectHacker
$\displaystyle n^2+(n+1)^2 = y^2$

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. - Sep 16th 2007, 08:30 AMadnan0