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Math Help - solving diophantine equations with modulo

  1. #1
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    Question solving diophantine equations with modulo

    For the Diophantine equation x^4+5y^3=2009, either find all integer solutions, or show that there are no integer solutions.

    How can I use modulo to solve this?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by jmedsy View Post
    For the Diophantine equation x^4+5y^3=2009, either find all integer solutions, or show that there are no integer solutions.

    How can I use modulo to solve this?
    Consider modulo 5.
    x^4=0,\ 1\ (mod\ 5)
    2009=4\ (mod\ 5)

    2009-x^4=4,\ 3\ (mod\ 5)
    2009-x^4=5y^3 has no integer solutions.
    x^4+5y^3=2009 has no integer solutions.
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  3. #3
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    Quote Originally Posted by alexmahone View Post
    Consider modulo 5.
    x^4=0,\ 1\ (mod\ 5)
    2009=4\ (mod\ 5)

    2009-x^4=4,\ 3\ (mod\ 5)
    2009-x^4=5y^3 has no integer solutions.
    x^4+5y^3=2009 has no integer solutions.
    That makes sense. Did you choose (mod 5) just because one of the terms including a variable was a multiple of 5?
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by jmedsy View Post
    That makes sense. Did you choose (mod 5) just because one of the terms including a variable was a multiple of 5?
    Yes.
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