Residues : The gateway to higher arithmetic I

TL;DR

This paper discusses the historical development of the concept of residues in number theory, highlighting Gauss's foundational role and how Dirichlet's interpretation shifted the paradigm towards higher arithmetic.

Contribution

It clarifies the historical evolution of residues, emphasizing Gauss's original paradigm shift and how Dirichlet's interpretation influenced subsequent mathematical approaches.

Findings

Gauss introduced residues as a fundamental concept in arithmetic.

Dirichlet's interpretation focused on remainders, obscuring Gauss's broader paradigm.

The shift from elementary to higher arithmetic was facilitated by these conceptual developments.

Abstract

Residues to a given modulus have been introduced to mathematics by Carl Friedrich Gauss with the definition of congruence in the `Disquisitiones Arithmeticae'. Their extraordinary properties provide the basis for a change of paradigm in arithmetic. By restricting residues to remainders left over by divison Peter Gustav Lejeune Dirichlet - Gauss's successor in G\"ottingen - eliminated in his `Lectures on number theory' the fertile concept of residues and attributed with the number-theoretic approach to residues for more than one and a half centuries to obscure Gauss's paradigm shift in mathematics from elementary to higher arithmetic.