# Thread: Disprove a statement

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.

2. Originally Posted by adnan0 Disprove the statement There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square.
Try n = 20: $\displaystyle 20^2 + (20 + 1)^2 = 400 + 441 = 841 = 29^2$.

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

3. Originally Posted by adnan0 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.)
$\displaystyle n^2+(n+1)^2 = y^2$
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.

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