Let be a Salem number of degree6for which for some . Then .

Let . Then is the minimal polynomial of a Salem number , forA > 0, by Lemma 2.1. Let denote the cyclotomic polynomial (the minimal polynomial of the primitive roots of unity), and define , so thatQis a nonreciprocal polynomial of degree32. 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 allnand if equality holds (i.e.,n = 8or17), then . The condition guarantees for alln.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 ofA, and hence the identity need only be established for two values ofA, e.g.A = 0andA = 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

33and not a proper divisor of33because of the positions of the two .