\documentclass{article}[11 pts] 

\begin{document}
\centerline{\bf THEORY OF NUMBERS, Math 115 A}
\centerline{\bf Homework 7 Due Wednesday November 27 2002}
\vskip 5pt

\begin{enumerate}

\item (not from the book) Prove or disprove
the following statement: For every odd number $n$, $\sigma (n) \leq 2n-1$.
HINT: Try to experiment with LOTS of numbers, use your intuition.

\item (7.1, 8) Show that there is no positive integer $n$ such that
$\phi(n)=14$.

\item (7.1, 18) Show that if $m,k$ are positive integers then
$\phi(m^k)=m^{k-1}\phi(m)$.

\item  (7.2, 12) Show that the equation $\sigma(n)=k$ has at most
a finite number of solutions when $k$ is a positive integer.

\item (7.2, 11) What is the product of the positive divisors of a
positive integer? Write a formula for it.

\item (7.3, 4) Find a factor of each of the following integers:
$2^111-1$, $2^{289}-1$, and $2^{46189}-1$.

\item (7.4, 6) Find all composite numbers $n$ between 100 and 200
with $\mu(n)=-1$.

\item (7.4, 8) Find the value of the Mertens function $M(n)$ for
$n=100$.

\item (7.4, 22) Let $n$ be a positive integer. Show that
$\Pi_{d \vert n} \mu (d)$ equals (1) -1 if $n$ is a prime,
(2) Zero if $n$ has a square factor, and (3) 1 if $n$ is square-free
and composite.

\end{enumerate}

\end{document}





