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Math Help - integer solutions for a straight line

  1. #1
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    integer solutions for a straight line

    Hi,

    I came across this..

    It is well known that the equation Ix + my + n = 0, with rational integral coefficients, has either no solution in rational integers
    or an infinite number of solutions.
    here http://www.ams.org/journals/tran/194...-0006739-2.pdf

    How is it possible to determine whether a straight line has rational integer solutions or not?

    5x^2 - y^2 + 20 = 0 has integer solutions at (11, 25), (29, 65), (199, 445)......

    It appears that there are no integer solutions for 5x^2 - y^2 + 45 = 0 but I don't know how to go about proving (or disproving) there are no integer solutions.

    I'd be grateful if someone could provide any help.

    Thanks
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  2. #2
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    Re: integer solutions for a straight line

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  3. #3
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    Re: integer solutions for a straight line

    Quote Originally Posted by earboth View Post
    Thanks earboth, I'll take a peek at my post in a while

    btw, there are two corrections to my OP

    1) hyperbola, not straight line

    2) x and y must be coprime

    thanks again
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