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 
)
![[Annotate]](/organics/icons/sannotate.gif){kind=icon}
![[Shownotes]](../gif/annotate/sshow-31.gif)