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# Are Salem numbers of degree 6 beta numbers?

In this section we describe the results of a computation of the beta expansions for the 11836 Salem numbers of degree 6 and trace at most 15. The complete expansion was obtained except in 80 cases. The distribution of the pairs for these numbers is quite remarkable. For the most part there is a great deal of regularity, and the values of m and p are quite small: for all but 199 of the numbers surveyed, both m < 1000 and p < 1000. For 9609 of the numbers (81% of the total) we have m = 1, but, in contrast to the degree-4 situation, larger values of m do occur with a certain regularity. However, among the remaining 199 cases, there are at least 79 for which and at least two for which .

The two known examples with are and . This might suggest that these are not beta numbers. However, there are some very large values of m and p among the cases where has been determined. The (current) record is attained for , which has . Other notable values are with , and with .

A sample of these numerical results is presented in two tables organized as follows. In Table 1 of the Supplement all quintuples with are listed, provided both m < 10000 and p < 10000. The complete table for is over 20 times longer than that presented here and has much the same general appearance. Table 2 of the Supplement gives the 36 values with for which . This includes 18 cases where only a lower bound for D = m+p is known.

Although there must be other factors involved, the argument of § 6 suggests that the size of is directly related to the size of . In particular, numbers with smaller discriminants should have larger orbits. The prediction of § 6.6 is that the orbit will be finite provided . We illustrate this in Table 3, where we list the values of and for the 11 Salem numbers with a = -3, b=-1, so . The omitted values of c correspond to reducible polynomials.

One might expect that the size of the period of might depend on arithmetic properties of or perhaps on the existence of approximate multiplicative relationships between the conjugates of with . If this were true, then one would expect powers of to behave in a manner similar to . A counterexample to this expectation is the with , where . Here, has with .

Exploring this theme further, we consider the powers of the smallest Salem number of degree 6, , which has . For , Table 4 gives for the minimal polynomial of , the values of m and p, and . The factored form of is given for the sake of interest, although the arithmetic properties of appear to play no role here. Table 4 (powers of )

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