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Thread: Show that there exist no positive integers ...

  1. #1
    MHF Contributor alexmahone's Avatar
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    Show that there exist no positive integers ...

    Show that there exist no positive integers $\displaystyle m$ and $\displaystyle n$ such that $\displaystyle m^2+n^2$ and $\displaystyle m^2-n^2$ are both perfect squares.
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    Re: Show that there exist no positive integers ...

    There's a theorem by Fermat: The equation $\displaystyle x^4-y^4=z^2$ is not solvable in nonzero integers.
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    MHF Contributor alexmahone's Avatar
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    Re: Show that there exist no positive integers ...

    Quote Originally Posted by melese View Post
    There's a theorem by Fermat: The equation $\displaystyle x^4-y^4=z^2$ is not solvable in nonzero integers.
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    So, $\displaystyle m^2+n^2=a^2$ and $\displaystyle m^2-n^2=b^2$.

    $\displaystyle (m^2+n^2)(m^2-n^2)=a^2b^2$

    $\displaystyle m^4-n^4=(ab)^2$, which has no solution.
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