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# Patterns and regularities in the tables

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.

### 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.

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