On k-regularity of sequences of valuations and last nonzero digits

2022/02/20 by Sobolewski, Bartosz
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2202.09922

Abstract

Let b ≥ 2 be an integer base with prime factors p₁, …, 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) = (f₁(n), …, 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 for b prime obtained 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 by certain ternary quadratic forms.

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