Suppose the beta expansion of 
is 
, so that
the characteristic polynomial is
Let 
be the minimal polynomial of
. If this is to divide 
, then the quotient must be of the form x + d for
some integer 
. By a result of Parry mentioned earlier, since -d is a root of
other than 
, it must satisfy |d| < 2 and hence 
. From (4.4), 
, and since 
by Parry's criterion, we
must have d = 1. Equating the coefficients of 
with those of (4.4) thus
shows that
The conditions 
for all n imply 
, 
and 
.
The conditions 
and 
imply 
and 
.
Finally, if 
, i.e., C = A + B - 1, then we must have 
,
that is 
; but 
implies 
, so in this case A = C
and hence B = 1.
To finish the proof, we must verify that Parry's criterion follows from the
inequalities in the statement of the proposition, the equation (4.5) defining 
for
, and the assumed periodicity of 
. This involves checking only a finite
number of cases: for example, to verify that 
, we
note that 
follows from 
, and that if 
, then A
= C and B = 1, so 
(always true here), and 
. The remaining cases are similar and left to the reader.
Finally, we must verify that the exact period is 6 and not a divisor of 6.
For example, if p=1, then (4.5) would imply that A=C and B=2. Then we find that
, so that 
and hence P is reducible. Similarly,
p=2 or p=3 lead to 
and the reducibility of P.
