Products of $p$-Adic Valuation Trees

TL;DR

This paper introduces $p$-adic valuation trees to visually analyze prime divisibility in sequences, focusing on trees generated by linear and polynomial functions, and explores their infinite branches and node valuations.

Contribution

It develops a method to construct and analyze $p$-adic valuation trees for sequences defined by linear and polynomial functions, revealing their structure and valuation properties.

Findings

Characterization of infinite branches in valuation trees

Descriptions of valuations at terminating nodes

Insights into the structure of $p$-adic valuation trees for polynomials

Abstract

The study of prime divisibility plays a crucial role in number theory. The $p$-adic valuation of a number is the highest power of a prime, $p$, that divides that number. Using this valuation, we construct $p$-adic valuation trees to visually represent the valuations of a sequence. We investigate how nodes split on trees generated by linear functions with rational coefficients, as well as those formed from a product of linear or lower degree polynomials. We describe the infinite branches of these polynomial trees and the valuations of their terminating nodes.