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Solving nonlinear congruences

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data

    Find an integer x that is a solution (only need one solution, not all solutions). If no solution exists, prove that no solution can exist.

    13x = 13 mod 50032 , with x > 1. Note that 5003 is prime.

    Here, = means "congruent to"

    2. Relevant equations

    Not sure. I can solve linear congruences without too much trouble, but I haven't seen a congruence of this form before, and the book that I'm using is pretty low on good examples.

    3. The attempt at a solution

    I don't have one. Looking for a bump in the right direction.
     
  2. jcsd
  3. Dec 13, 2011 #2
    I spent a lot of time last night trying a bunch of different techniques on your problem and couldn't get much of anywhere. But since no one else has responded to it yet I will give you what little I have, which is just Fermat's Little Theorem:

    13^5003 = 13 mod 5003. But from here I don't know where to go.
     
  4. Dec 13, 2011 #3

    micromass

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  5. Dec 13, 2011 #4
    I can use Euler's theorem since 13 and 50032 are coprime.

    However, I then end up with: 13phi(50032) = 1 mod 50032. How can I manipulate this so that I have 13 mod 50032 instead of 1 mod 50032?
     
  6. Dec 13, 2011 #5
    Can't you just multiply both sides by 13?
     
  7. Dec 13, 2011 #6
    Sigh .... yes. Yes I can. I think it's time for a break, I've been staring at this for too long and I'm starting to forget about basic math.

    Thanks for the wake up call!
     
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