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Solution to the general congruence

  1. Aug 6, 2006 #1
    :rofl: :confused: :cool: :rolleyes: Can anyone help me to provide a solution to the general congruence:

    [tex] x^n =a Mod (b) [/tex] a,n and b integers or the integer solution to

    equations of the form:

    [tex] a x^n + by= c [/tex] solutions for integer x and y :grumpy: :grumpy:
     
  2. jcsd
  3. Aug 6, 2006 #2
    what prior knowledge was given in the course? Roots? or Powers n^x=amodb? P
     
  4. Aug 6, 2006 #3

    Hurkyl

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    Plug it into Magma. :smile:


    I would do it as follows.

    (1) First, find the prime factorization of b. Let's assume b = p^2 * q

    (2) Find the n-th root of a modulo p and modulo q. (I know the Shanks-Tonelli algorithm works for square roots, and can be adapted for arbitrary roots. There may be a better way)

    (3) Use Hensel lifting to find an n-th root of a modulo p^2

    (4) Use the Chinese Remainder Theorem to find an n-th root of a modulo b
     
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