Differential calculus with integers

TL;DR

This paper surveys the arithmetic analogue of differential calculus, replacing functions with numbers and derivatives with Fermat quotient operators, discussing motivations, constructions, results, applications, and open problems.

Contribution

It provides a comprehensive overview of the development and current state of differential calculus with integers, highlighting key concepts and open questions.

Findings

Establishes the analogy between differential equations and arithmetic functions

Summarizes main results and applications of the theory

Identifies open problems and future research directions

Abstract

Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions, results, applications, and open problems of the theory.