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Math Help - Multiplicative Function Questions

  1. #1
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    Multiplicative Function Questions

    1) Show that if p is prime, 2^ap+1 is composite for a = 1,2,....,r and p is not a Fermat prime, where r is a positive integer, then \phi(n) = 2^rp has no solution.

    2) The arithmetic funtion g is said to be the inverse of the arithmetic function f if f * g = g * f = i. Show that the arithmetic function f has an inverse if and only if f(1) does not equal 0. Show that if f has an inverse it is unique.
    (Hint: When f(1) is not equal to 0, find the inverse f^{-1} of f by calculating f^{-1}(n) recursively, using the fact that i(n) = summation (f(d)f^{-1}(n/d).)
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  2. #2
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    1) seems like it shouldn't be too difficult. I don't get what I'm trying to prove though. How would the value of 2^rp be able to show that it is not solvable? I know \phi(n) is always even when n \geq 2. But 2^rp is always going to be even, so I'm confused as to how I prove this.

    2) I don't even understand how I recursively go back and do this.

    f*f{-1} = i, and set i = summation....
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