### Proposition 4.3

Let be a Salem number of degree 6 for which

for some . Then .

### Proof

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 .