\problemdata       % Takes 5 *mandatory* arguments
  {arithmetic-expr}        % latex-friendly label for the prob.
  {induction}              % The topic(s) of the problem content.
  {}                       % Source (if known)
  {Never?}                 % Usage (list ones you are aware of).
  {Nick Matsakis, S02}     % Last revision info (author, date).
%% Original problem, recursively defined set (of arithmetic expressions);
%% modular arithmetic.
\begin{problem}
Define the set of \emph{Arithmetic Expressions} (AExp's) recursively as follows:
\begin{enumerate}

\item base cases:
\begin{enumerate}

\item a \emph{variable} is an AExp.

\item an \emph{integer} is an AExp.

\end{enumerate}
\item if $F,G$ are AExp's, the ($F$ + $G$) is an AExp called a \emph{sum}.

\item if $F,G$ are AExp's, then ($F \cdot G$) is an AExp called a \emph{product}.

\end{enumerate}
For example,
\[
(((3 \cdot x) + y) \cdot ((y\cdot (-5)) + 92)).
\]
Omitting unnecessary parentheses and the multiplication sign ($\cdot$), we
could write this in the easier to read form
\begin{equation}\label{ae1}
(3 x + y)(y(-5) + 92).
\end{equation}

The value of expression~\ref{ae1} depends on the values of $x$ and $y$ in
the obvious way.  For example, if $x=2$ and $y=1$, its value is $7\cdot87
=609$.
\begin{problemparts}

\problempart Prove that for any AExp, $E$, with variables $x_1, x_2,\dots$
and any $n>1$, if $k_1 \equiv l_1 \mod n$, $k_2 \equiv l_2 \mod n$, \dots,
the value of $E$ when the variables equal $k_1,k_2,\dots$ is $\equiv \mod
n$ to the value of $E$ when the variables equal $l_1,l_2,\dots$.

\problempart If an AExp is not constant, then the value of the expression
at some natural number assignment to its variables is not a prime.

\end{problemparts}
\end{problem}
