Investigations on the properties of the arithmetic derivative

TL;DR

This paper explores the properties of arithmetic differentiation, extending the concept to rational numbers, and presents new theorems that deepen understanding of its mathematical implications and connections to prime number theory.

Contribution

It introduces new theorems related to arithmetic differentiation of rational numbers, expanding the scope of the field and its potential applications.

Findings

New theorems on arithmetic differentiation of rational numbers

Insights into the relationship between arithmetic differentiation and prime number conjectures

Enhanced understanding of the mathematical structure of arithmetic derivatives

Abstract

We investigate the properties of arithmetic differentiation, an attempt to adapt the notion of differentiation to the integers by preserving the Leibniz rule, (ab)' = a'b + ab'. This has proved to be a very rich topic with many different aspects and implications to other fields of mathematics and specifically to various unproven conjectures in additive prime number theory. Our paper consists of a self-contaited introduction to the topic, along with a couple of new theorems, several of them related to arithmetic differentiation of rational numbers, a topic almost unexplored until now.