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Math Help - FLT Problem

  1. #1
    Junior Member
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    May 2009
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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 x = 5n;y = 3n then you can check that w=4n. And on the other hand we have: 1 = {z^2} - 34{n^2} which is a Pell's Equation. (And there are infinitely many solutions)


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

    For example, the next is: (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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