Contents
Next: The computation of
Up: No Title
Previous: Are Salem numbers
![[Annotate]](/organics/icons/sannotate.gif)
![[Shownotes]](../gif/annotate/shide-41.gif)
Many regularities are apparent in Table 1. For example, it is easy to spot numerous
occurences of 
, 
, 
, 
, etc., and even to guess the
general pattern of such occurences. Such results can usually be proved by the method
of [3]. That is, one guesses the expansion 
of 
, verifies that this purported expansion is a legitimate expansion of
a beta number 
according to Parry's criterion, computes the characteristic
polynomial 
of 
by (1.2), and verifies that 
is divisible by 
,
the minimal polynomial of 
. For example, here are some simple cases:
Proposition 4.1
Let 
be a Salem number of degree 6 with minimal
polynomial 
given by (2.1). Then 
if and only if
and 
and 
.
[Proof]
Proposition 4.2
Let 
be a Salem number of degree 6 with minimal
polynomial 
given by (2.1). Then 
if and only if
, where A,B and C are positive integers satisfying
, 
, 
with the further condition that if
A + B - 1 = C, then A = C and B = 1.
[Proof]
Proposition 4.3
Let 
be a Salem number of degree 6 for which
for some 
. Then 
.
[Proof]
Remark 4.1 The condition in Proposition 4.3 is only a sufficient condition
for 
. There are many other occurrences of 
besides the ones
described here.
Contents
Next: The computation of
Up: No Title
Previous: Are Salem numbers