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Math Help - Rational solutions to an equation

  1. #1
    Senior Member Pinkk's Avatar
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    Rational solutions to an equation

    Prove that the equation x^{2} + y^{2} = 7 has no rational solutions.

    I tried to go about this using the fact that for any integer a, a^{2} \equiv 0(mod 4) or a^{2} \equiv 1(mod4), but when I supposed that there existed a rational solution (\frac{p}{q},\frac{m}{n}) and arrived to the equation (pn)^{2} + (qm)^{2} = 7(qn)^{2}, I can't draw the conclusion I want. How do I go about this? Any help would be appreciated.
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  2. #2
    Senior Member Pinkk's Avatar
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    Re: Rational solutions to an equation

    I used mod7 instead, and I reached the conclusion that pn, qm, qn are all divisible by 7. I know this in turn means the fractions assumed to be solutions are irreducible but I don't know how to prove that exactly.
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