On the other hand, if is an algebraic integer, then all lie in . Specifically, if has degree d and minimal polynomial P, and if is the polynomial of degree d-1 definied by , where is as in § 1, then . The thus provide an exact representation of , so the question can be effectively answered. Since the satisfy the recurrencethe computation simply involves a shift of the coefficients of followed by the replacement of by .
The only difficulty is the determination of . This can be computed using an approximation to of modest accuracy unless is unusually close to an integer c. If is a Salem number, standard arguments from transcendence theory show thatwhere denotes the sum of the absolute values of the coefficients of . One simply observes that there is an obvious bound on the conjugates of owing to the fact that the other conjugates of all have absolute value , and that the product of all the conjugates is a nonzero integer.
In practice, we can simply choose an approximation to of fixed accuracy . (We usually chose to be or .) The number is an approximation to whose accuracy is easily estimated. If , and if , then . Thus, if the distance from to the nearest integer is at least , then .
Our algorithm begins with (double precision). If at some point in the computation, the criterion of the previous paragraph fails, the computation of is repeated with a having accuracy (quadruple precision). If the criterion fails at this level of accuracy, then the computation is terminated. This occurred for only five values of in our entire project, one of which can be recognized in Table 2: is not given, but the corresponding lower bound on D is . For each , the values of n for which a change from double precision to quadruple precision was necessary were recorded in a file during the computation. We indicate below how these values were used in some cases to determine some of the larger values of .
As a sample of the accuracy needed, suppose we were to take for the with . Then for n = 167305 one computes , suggesting that , while in fact , so . On the other hand, for most values of n, a far less accurate value of would suffice. The two-tier arrangement described in the previous paragraph allows one to make use of the fast double-precision multiplication on the machine used (an Amdahl 5860) while retaining the benefits of a higher-precision calculation when required. One could easily envision a multiple-tier approach. The sequence is periodic if and only if is bounded. Thus, if is a beta number, its expansion can be computed using an approximation of some fixed accuracy (which depends on , of course).