Let be a Salem number of degree6with minimal polynomial given by (2.1). Then if and only if and and .

For convenience, write . The minimal polynomial of must satisfy This implies that Next we must determine the conditions under which the expansion satisfies , for alln, and that, for alln > 1, where, in (4.3),>denotes lexicographical order [7],p.407. From (4.2), we see that the condition , for alln, implies , , and . But it is easily seen that these conditions insure that (4.3) holds for allnand hence that is the beta expansion for a number given by (1.3).The right member of (4.2) defines a sequence with

m = 1andpa divisor of5, so eitherp = 1or5. However,p = 1occurs only ifB = C = A-1and this is the expansion of a beta with minimal polynomial , not a Salem number. This can be easily seen from (1.3) or by observing that the polynomial vanishes atx = A+1.The periodicity of shows that is the characteristic polynomial of . But (4.1) shows that , the minimal polynomial of . Hence, , so has .