Thread: Multiplicative Function Questions

1) Show that if p is prime, $\displaystyle 2^ap+1$ is composite for a = 1,2,....,r and p is not a Fermat prime, where r is a positive integer, then $\displaystyle \phi(n) = 2^rp$ 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 $\displaystyle f^{-1}$ of f by calculating $\displaystyle f^{-1}(n)$ recursively, using the fact that i(n) = summation $\displaystyle (f(d)f^{-1}(n/d)$.)

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 $\displaystyle 2^rp$ be able to show that it is not solvable? I know $\displaystyle \phi(n)$ is always even when $\displaystyle n \geq 2$. But $\displaystyle 2^rp$ 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....