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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

, verifies that this purported expansion is a legitimate expansion of a beta number

according to Parry's criterion, computes the characteristic polynomial

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.