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\noindent \textbf{Math 565 - Fall 2019 \\
 Homework 6} \\
Due October 28th, 2019 \vspace{-4mm}\\

\noindent \small \textit{I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply our notes of our observations. This view has been held, in one form or another, by many philosophers of high reputation from Plato onwards, and I shall use the language which is natural to a man who holds it.
}\\\vspace{1mm} \hfill--- G. H. Hardy


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Problem 6.4-11 from the book and the following.
\begin{enumerate}
\item Suppose $f(n) = \frac{n-1}{n+1}$ and $g(n)$ is a function with the property that \[f(n) = \sum_{d|n} g(d).\]
Compute $g(24)$.
\item For which prime $p<20$ is there the most primitive roots? (Hint: don't actually find the primitive roots...)
\item Using the information that 7 is a primitive root modulo 22, determine all
integers modulo 22 that are primitive roots.
\item \begin{enumerate} \item Determine the order of 2 modulo $2^n -1$.
\item Use this to prove that $\varphi(2^n- 1)$ is divisible by $n$.
\end{enumerate}


\end{enumerate}

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