### Proposition 4.2

Let be a Salem number of degree 6 with minimal polynomial given by (2.1). Then if and only if , where A,B and C are positive integers satisfying , , with the further condition that if A + B - 1 = C, then A = C and B = 1.

### Proof

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.