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\centerline{\BF Additional Exercises}
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\noindent
{\bf 1.} Suppose that $p$ and $q$ are relatively prime positive integers.
Show that if $\cos p\alpha$, $\cos q\alpha$, and $\cos (p+q)\alpha$ are
all rational, then $\cos \alpha$ is rational.
\bigskip
\noindent\llap{\bf *}%
{\bf 2.} Suppose that $p$ and $q$ are relatively prime positive integers.
Show that if $\cos p\alpha$ and $\cos q\alpha$ are rational, then $\cos\alpha$
is rational or $\alpha$ is a multiple of $\pi/6$.

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\noindent
{\bf 3.} Suppose we extend the definition of divisibility as follows:
Let $m$ be a non-zero integer, and let $a/b$ be a rational number in
lowest terms, i.e., $(a, b) = 1$.  We say that $m$ divides $a/b$, and write
$m | a/b$, if $(m, b) = 1$ and $m|a$.  Suppose also that for rational numbers
$r$, $s$ with denominators relatively prime to $m$, we say that
$r\equiv s\pmod m$ if $m|(r-s)$.\hfil\break
(a) Let $r$, $R$, $s$, $S$ be rational numbers whose denominators are 
relatively prime to $m$.  Show that if $r\equiv R\pmod m$ and 
$s\equiv S\pmod m$, then $r+s\equiv R+S\pmod m$ and $rs\equiv RS\pmod m$.
\hfil\break
(b) Suppose that $r= a/b$ is a rational number with $(b, m) = 1$.  Choose an
integer $\overline{b}$ such that $b\overline{b}\equiv 1\pmod m$.  Show that
$r\equiv a\overline b\pmod m$.  Show that if $c$ is an integer such that
$r\equiv c\pmod m$, then $c\equiv a\overline b\pmod m$.\hfil\break
(c) Suppose that $a$, $A$, $b$, $B$ are integers such that $a\equiv A\pmod m$,
$b \equiv B\pmod m$, and $(b, m) = 1$.  Show that $a/b \equiv A/B\pmod m$.

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{\bf 4.} Let $f(z) = \sum_{i=0}^\infty a_iz^i$ and 
$g(z) = \sum_{i=0}^\infty b_iz^i$ be two formal power series with integral 
coefficients.  (By ``formal'' we mean that we don't care whether the radius
of convergence is positive.)  Define $f\equiv g\pmod m$ if 
$a_i \equiv b_i\pmod m$ for all $i = 0, 1, \ldots$.\hfil\break
(a) Suppose that $f$, $F$, $g$, $G$ are power series with integral 
coefficients.  Show that if $f\equiv F \pmod m$ and $g\equiv G \pmod m$, then
$f+g\equiv F+G\pmod m$ and $fg\equiv FG\pmod m$.\hfil\break
(b) Suppose that $f$ and $F$ are power series with integral coefficients, both
of which have constant term 0.  Show that if $f\equiv F\pmod m$, then
$$
\frac1{1+f}\equiv \frac1{1+F}\pmod m.
$$

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{\bf 5. }Let $A$ be an $n\times n$ matrix with integral elements.  Show that
if $(\det(A), m)=1$, then for any $(c_1, c_2, \ldots, c_n)\in \bold Z^n$ there
is a unique $n$-tuple $(x_1, x_2, \ldots, x_n)$ of residue classes (mod $m$)
such that
$$
\sum_{j=1}^n a_{ij}x_j \equiv c_j\pmod m
\tag $\ast$
$$
for $i = 1, 2, \ldots, n$.  Also, show conversely that if $(\det(A), m) > 1$, 
then the number of solutions of ($\ast$) is either 0 or $> 1$, depending on
the choice of $(c_1, c_2, \ldots, c_n)$.

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\noindent
{\bf 6.} We recall that the binomial coefficient $\big({x\atop k}\big)$ is a polynomail
in $x$, namely $\big({x\atop k}\big)= x(x-1)\cdots(x-k+1)/k!$.  Thus 
$\big({-1/3\atop k}\big)$ is a rational number, say
$$
\Big({-1/3\atop k}\Big)= \frac{a_k}{q_k}
$$
where $(a_k,q_k)=1$ and $q_k > 0$.  \hfill\break
(a) Show that $q_k$ is a power of $3$.\hfill\break
(b) Show that $a_k$ is odd if and only if $k$ can be written as a sum of distinct 
powers of $4$.
\headline{ \hfil \bf Additional exercises\hfil}
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\noindent
{\bf 7.} (Nicol--Selfridge) Show that if $n$ is an odd integer $> 3$, then 
there is a prime number $p$ such that $p\nmid n$ and $p|(2^{\varphi(n)}-1)$.

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