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Math Help - Show a^n = 1 (mod 2^n+2) if a odd.

  1. #1
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    Show a^n = 1 (mod 2^n+2) if a odd.

    I know I have to use induction. The n=1 is simple:
    a^2 = (2j+1)^2 = 4j^2 + 4j + 1 = 1 mod 8 = 1 mod 2^3

    Then we assume for a^n that the claim is true.

    However I'm not sure on the inductive step.

    This is a HW assignment and help would be much appreciated. Thanks!
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  2. #2
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    Re: Show a^n = 1 (mod 2^n+2) if a odd.

    SOLVED:

    Note a^n -1 = [(a^n-1) - 1][(a^n-1) + 1]
    The left term is divisible by 2^n+2 by inductino and the right term is divisible by 2. Thus their product is divisible by 2^n+3.
    Thanks from jakncoke
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