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Math Help - Disprove a statement

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    Disprove a statement

    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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    Quote Originally Posted by adnan0 View Post
    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
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    Quote Originally Posted by adnan0 View Post
    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.
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    Quote Originally Posted by topsquark View Post
    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.
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