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Math Help - Number Theory Question (Mobius Theorem)

  1. #1
    Senior Member
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    Number Theory Question (Mobius Theorem)



    Hi guys, I just have a few problems with part b) and c), I can do part a) the following is my working but I'm just not sure how to solve b) and c). Any help would be appreciated, cheers!

    If we let \displaystyle{f(d) = d^k} and \displaystyle{F(n) = \sigma_k (n)} then since \displaystyle{\sigma_k(n) = \sum_{d|n} d^k} we have \displaystyle{F(n) = \sum_{d|n} f(d)}

    Using the Mobius Inversion Theorem yields:

    \displaystyle{f(n) = \sum_{d|n} \mu(d) F\left(\frac{n}{d}\right)}

    \displaystyle{\Rightarrow n^k = \sum_{d|n} \mu (d) \sigma_k \left(\frac{n}{d}\right)} as required.
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  2. #2
    Super Member PaulRS's Avatar
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    For (b) remember Fermat's Little Theorem, we have a^p\equiv a (\bmod.p) for all a\in \mathbb{Z}
    For (c) rewite it as \sigma_k ( n) = n^k + 1, what are the divisors?
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