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Math Help - congruence modulo 2^n

  1. #1
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    congruence modulo 2^n

    Let x be an ODD integer show that for n>=3
    x^2^(n-2) is congruent to 1 (mod 2^n)

    Sorry about the superscripts I dont use the computer for math often.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    It holds for n=3; I'll let you prove that. Suppose it holds up to n-1. Note that x^{2^{n-2}}-1=(x^{2^{n-3}}-1)(x^{2^{n-3}}+1); and by the induction hypothesis, x^{2^{n-3}}-1 is divisible by 2^{n-1}, and since x is odd, x^{2^{n-3}}+1 is divisible by 2 so (x^{2^{n-3}}-1)(x^{2^{n-3}}+1)=x^{2^{n-2}}-1 is divisible by 2^n.
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