Algorithm to Detect Periodicity by Interleaving Sequences

Abstract

We define an algorithm which begins with an sequence of sequences, and produces a single sequence, with following property: If at least one of the original sequences has a tail that is periodic, then the output sequence has a periodic tail, and conversely. Our purpose is to supplement a result by Dasaratha et al., in which a real number input results in a countable family of sequences, with the property that at least one is eventually periodic if and only if the input is a cubic irrational number. This result, in the context of that one, attempts to address Hermite's problem, which asks for some generalization of continued fractions that can detect cubic numbers via periodicity. We note in our ending remarks how this particular trick fails to give a satisfying resolution to Hermite's problem, but we present it anyway, for anyone whom it may interest.