A Salem number of degree 6 has a minimal polynomial of the following form:

where a,b and c are integers. The trace of is , where the sum is over the six conjugates of , i.e., the six roots of (2.1). Two of these conjugates are and and the remaining conjugates satisfy , so a bound on |a| implies a bound on and hence on |b| and |c|. So, for fixed a, there is a finite set of with . Observe that

so for all Salem numbers of degree 6. In fact, there are no such numbers with trace -1, 4 with trace 0, 15 with trace 1, and 39 with trace 2.
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