1. Disprove a statement

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

Disprove the statement “There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square”.
Try n = 20: $20^2 + (20 + 1)^2 = 400 + 441 = 841 = 29^2$.

That's it for n less than 100. Probably aren't any more out there, but I'll leave that for someone else to prove (who knows how to do it.)

-Dan

Disprove the statement “There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square”.
The negation of this statement would be. "There exists a positive integer (in fact infinitely many, but that is something else) n>3 so that n^2+(n+1)^2 is a perfect square" So find one.

4. Originally Posted by topsquark
Probably aren't any more out there, but I'll leave that for someone else to prove (who knows how to do it.)
$n^2+(n+1)^2 = y^2$
Thus,
$2n^2+2n+(1-y^2)=0$
We require the distriminant to be a square.
$4-2(1-y^2) = x^2$
$4-2+2y^2 = x^2$
$x^2-2y^2 = 2$.
This is a Pellain equation-look-a-like.
If it has 1 solution it has infinitely many.
Indeed it does.