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Thread: FLT Problem

  1. #1
    Junior Member
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    FLT Problem

    Question:

    Prove the following assertion: The system of simultaneous equations x^2+y^2=z^2-1 and x^2-y^2=w^2 has infinitely many solutions in positive integers x, y, z, w.

    I was thinking I could consider that for any integer n>=1 and then take x=2n^2 and y=2n, but I am not sure how that would work out.
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  2. #2
    Super Member PaulRS's Avatar
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    Take $\displaystyle x = 5n;y = 3n$ then you can check that $\displaystyle w=4n$. And on the other hand we have: $\displaystyle 1 = {z^2} - 34{n^2}$ which is a Pell's Equation. (And there are infinitely many solutions)


    The minimal solution is $\displaystyle (z,n)=(35,6)$ and the solutions are generated by: $\displaystyle \left\{ \begin{gathered}
    {z_k} = \tfrac{{{{\left( {35 + \sqrt {34} \cdot 6} \right)}^k} + {{\left( {35 - \sqrt {34} \cdot 6} \right)}^k}}}
    {2} \hfill \\
    {n_k} = \tfrac{{{{\left( {35 + \sqrt {34} \cdot 6} \right)}^k} - {{\left( {35 - \sqrt {34} \cdot 6} \right)}^k}}}
    {{2\sqrt {34} }} \hfill \\
    \end{gathered} \right.
    $

    For example, the next is: $\displaystyle (z_2,n_2)=(2449,420)$
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  3. #3
    Junior Member
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    Thanks for your help, however, I have not yet learned about Pell's Equation in my class. Is there another way to solve this?
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