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 $\displaystyle (f(d)f^{-1}(n/d)$.)