Disprove a statement

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September 16th 2007, 07:41 AM
adnan0

Disprove a statement

Disprove the statement “There is no positive integer n > 3 such that n^2 + (n + 1)^2 is a perfect square”.
September 16th 2007, 08:16 AM
topsquark

Quote:

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: $20^2 + (20 + 1)^2 = 400 + 441 = 841 = 29^2$ .

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
September 16th 2007, 08:17 AM
ThePerfectHacker

Quote:

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.
September 16th 2007, 08:19 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.
September 16th 2007, 08:30 AM
adnan0

Thanks ThePerfectHacker ..
i alrady knew that 20 is one of the solution .. will work on

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