Proposition 4.1

Let be a Salem number of degree 6 with minimal polynomial given by (2.1). Then if and only if and and .


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 .