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Math Help - Diopantine equation #2: y^2+y+1=x^2.

  1. #1
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    Diopantine equation #2: y^2+y+1=x^2.

    Show that y^2+y+1=x^2 has no positive integer solution.
    Last edited by mr fantastic; January 11th 2011 at 11:28 AM.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by dila View Post
    Show that y^2+y+1=x^2 has no positive integer solution.
    x^2=y(y+1)+1

    Clearly, x^2 is odd.

    \implies x is odd.

    \implies x^2-1 is a multiple of 8.

    \implies y(y+1) is a multiple of 8.

    \implies y\ or\ y+1 is a multiple of 8.

    Can you proceed? (because I can't!)
    Last edited by alexmahone; January 11th 2011 at 01:05 AM.
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  3. #3
    MHF Contributor alexmahone's Avatar
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    New solution:

    y^2<y^2+y+1<(y+1)^2\implies y^2<x^2<(y+1)^2, which implies that there is a perfect square between two consecutive perfect squares, which is impossible.
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