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Math Help - Quadratic Diophantine equation

  1. #1
    Junior Member Greg98's Avatar
    Joined
    Oct 2009
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    Brugge, BEL
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    Quadratic Diophantine equation

    Hello,
    the task is to solve (find all integer solutions) following quadratic Diophantine equation:
    x^2-4y^2=65

    I think I need to use modular arithmetic for obtaining solutions. Maybe I should start investigating squares of integers mod \ x, x = 1,2,...

    Also I peeled answer from WolframAlpha, but that doesn't give me any hint how to get it myself. WolframAlpha gave the following solutions:
    x = \pm 33, \ y = \pm 16
    x = \pm 9, \ y = \pm 2

    So, any help is appreciated. Thanks!
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  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
    Posts
    485
    I recommend factoring the left hand side:

    (x+2y)(x-2y)=65

    Now the terms on the left must be factors of 65. There are several choices: 1 and 65, 65 and 1, 5 and 13, 13 and 5 (and all of the appropriate negative pairs). For example,

    x+2y=1
    x-2y=65

    Solving the system gives x=33,y=-16. Try it for all of the other pairs and you will get your entire solution set.

    Good luck.
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