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A Find positive integer solutions to a/(b+c)+b/(a+c)+c/(a+b)=4

  1. Aug 6, 2017 #1

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    What an innocently looking equation.
    If we allow negative integers, a=4, b=-1, c=11 is a solution.

    Do some tricks with divisibility?
    Solve for a?
    Brute force with the computer?


    It won't help. There are solutions, but the smallest solution has 80 digits.

    What happens if we replace 4 by other integers?
    a/(b+c)+b/(a+c)+c/(a+b)=178?
    There are integer solutions, but the smallest one has nearly 400 million digits.

    A great example how simple looking problems can have very complicated solutions.

    Paper: An unusual cubic representation problem (PDF)
    Here is a discussion
     
  2. jcsd
  3. Aug 6, 2017 #2

    fresh_42

    Staff: Mentor

    I thought Andrew had already demonstrated it.
     
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