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\handout{ps5}{Problem Set 5}{24 October 1998}

\due Monday, 2 November.

\medskip

\problem Prove that no nonconstant multivariate Diophantine
polynomial\footnote{Saying a polynomial is ``Diophantine'' means it has
integer coefficients and its variables range over only the integers.} can
have a range consisting entirely of various powers of two.

\problem Prove that if an integer in the range of a nonconstant
multivariate Diophantine polynomial is divisible by some integer $k$, then
there are infinitely many integers in the range which are divisible by
$k$.  Conclude that no nonconstant multivariate Diophantine polynomial can
have a range consisting entirely of various primes.

\problem Extend the Scheme Substitution Model in the file \texttt{SM.SCM}
(available on the course website) to include some the following Scheme
forms: \texttt{cond} as a ``first-class'' form -- not sugar;
\texttt{let*};  lambda expressions taking a variable number of arguments, e.g.,
\begin{lisp}
(lambda l 
  (if (null? l)
      (begin (display ``I need numerical arguments'') 'uh-oh)
      (apply + l)))
\end{lisp}
(Optional: add \emph{internal} \texttt{define}'s, possibly as sugar but
better as a first-class form.)

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