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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 .

### 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 .

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.

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