Number Theory Question (Mobius Theorem)

**usagi_killer** · March 7th 2011, 02:16 AM

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.

**PaulRS** · March 7th 2011, 02:28 AM

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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