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

TL;DR

This paper classifies the $k$-regularity of sequences of valuations and last nonzero digits in $b$-adic expansions of $p_i$-adic analytic functions, extending previous results and applying to Lucas sequences and quadratic form representations.

Contribution

It provides a complete classification of $k$-regularity for these valuation sequences, generalizing prior prime-base results and linking to quadratic form representations.

Findings

Classified $k$-regularity for sequences of valuations and last nonzero digits.

Extended results from prime to composite bases with multiple prime factors.

Applied to Lucas sequences and quadratic form representations.

Abstract

Let $b \geq 2$ 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 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.