Legendre's formula and $p$-adic analysis

TL;DR

This paper explores the relationship between Legendre's formula for p-adic valuations and p-adic weights, analyzing their increments and proposing an arithmetic framework within p-adic analysis.

Contribution

It introduces a detailed examination of the connection between p-adic valuations and weights, and proposes an arithmetic approach to their increments in p-adic analysis.

Findings

Relationship between p-adic valuation and p-adic weight clarified

Analysis of increments in p-adic valuations and weights

Proposed arithmetic framework for p-adic increments

Abstract

In number theory, we know Legendre's formula $ v_p(n!) = \sum_{k \ge 1} \lfloor \frac{n}{p^k} \rfloor $, which calculates the $p$-adic valuation of the factorial, i.e. the exponent of the greatest power of a prime $p$ that divides $n!$. There is also the second (or alternative) equality $ v_p (n!) = \frac{n-s_p(n)}{p-1} $ where $s_p(n)$ is the $p$-adic weight of $n$ or the sum of digits of $n$ in base $p$. Both kinds of Legendre's formula allow us to determine valuations of the natural number, the odd factorial, binomial coefficients, Catalan numbers, and other combinatorial objects. The article examines the relationship between the $p$-adic valuation and $p$-adic weight and considers their increments. The arithmetic of the $p$-adic increments is proposed.