For convenience, write 
. The minimal polynomial
of 
must satisfy
This implies that
Next we must determine the conditions under which the expansion satisfies 
,
for all n, and that, for all n > 1,
where, in (4.3), > denotes lexicographical order [7],p.407. From (4.2), we see
that the condition 
, for all n, implies 
, 
,
and 
. But it is easily seen that these conditions insure that (4.3) holds
for all n and hence that 
is the beta expansion for a number 
given by (1.3).
The right member of (4.2) defines a sequence with m = 1 and p a divisor of 5, so
either p = 1 or 5. However, p = 1 occurs only if B = C = A-1 and this is the
expansion of a beta with minimal polynomial 
, not a Salem number.
This can be easily seen from (1.3) or by observing that the polynomial
vanishes at x = A+1.
The periodicity of 
shows that 
is
the characteristic polynomial of 
. But (4.1) shows that 
, the
minimal polynomial of 
. Hence, 
, so 
has 
.
