Let 
. Then 
is the minimal
polynomial of a Salem number 
, for A > 0, by Lemma 2.1. Let 
denote
the 
cyclotomic polynomial (the minimal polynomial of the primitive
roots of unity), and define 
, so that Q is a nonreciprocal polynomial of degree 32. We claim that the beta
expansion of 
has 
and is given explicitly by
To prove this, we note that, for 
, the sequence 
defined by (4.6) is
the beta expansion of some 
, since it satisfies
Parry's criterion. This is obvious since
for all n and if equality holds (i.e., n = 8 or 17), then 
. The condition 
guarantees 
for all n.
Let 
be the characteristic polynomial of 
. We must next
verify that
This is an elementary algebraic computation
which is made easier by observing that the coefficients of 
and 
are
linear functions of A, and hence the identity need only be established for two values of
A, e.g. A = 0 and A = 1. We leave this to the reader who may wish to use a
computer algebra system.
Since 
has no positive roots, (4.7) implies that 
, and
hence that (4.6) is the beta expansion of 
. It is clear in this case
that the period of the expansion is 33 and not a proper divisor of 33
because of the positions of the two 
.
