On k-regularity of sequences of valuations and last non-zero digits

Abstract

Let b ge 2 be an integer base with prime factors p_1, ldots , p_s. In this paper we study sequences of “b-adic valuations” and last nonzero digits in b-adic expansions of the values f(n) = (f_1(n), ldots , f_s(n)), where each f_i is a p_i-adic analytic function. We give a complete classification concerning k-regularity of these sequences, which generalizes a result obtained for b prime by Shu and Yao. As an application, we strengthen a theorem by Murru and Sanna on b-adic valuations of Lucas sequences of the first kind. Moreover, we derive a method to determine precisely which terms of these sequences can be represented as a sum of three squares.