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In this section we describe the results of a computation of the beta expansions for the

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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