# Disprove a statement

• Sep 16th 2007, 08:41 AM
Disprove 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, 09:16 AM
topsquark
Quote:

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
• Sep 16th 2007, 09:17 AM
ThePerfectHacker
Quote:

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.
• Sep 16th 2007, 09:19 AM
ThePerfectHacker
Quote:

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.
• Sep 16th 2007, 09:30 AM