The Arithmetic Partial Derivative

TL;DR

This paper studies the properties of the arithmetic partial derivative, proving periodicity in its higher derivatives' p-adic valuations and providing criteria for integrality of anti-partial derivatives, with applications to counting integers with a fixed number of such derivatives.

Contribution

It introduces new results on the periodicity of p-adic valuations of higher derivatives and criteria for integrality of anti-partial derivatives, expanding understanding of this arithmetic operator.

Findings

p-adic valuation of higher derivatives is eventually periodic

Criteria established for when integers have integral anti-partial derivatives

Infinitely many integers have exactly n integral anti-partial derivatives for any n

Abstract

The arithmetic partial derivative (with respect to a prime $p$) is a function from the set of integers that sends $p$ to 1 and satisfies the Leibniz rule. In this paper, we prove that the $p$-adic valuation of the sequence of higher order partial derivatives is eventually periodic. We also prove a criterion to determine when an integer has integral anti-partial derivatives. As an application, we show that there are infinitely many integers with exactly $n$ integral anti-partial derivatives for any nonnegative integer $n$.