Research on Arithmetic

TL;DR

Lagrange's 1773 work systematically analyzed numbers represented by quadratic forms, establishing theorems on their divisors and developing methods to classify their possible forms, foundational to number theory.

Contribution

First systematic analysis of numbers represented by quadratic forms, including divisor theorems and methods to determine their possible forms, pioneering work in algebraic number theory.

Findings

Proved theorems on divisors of quadratic form numbers

Developed a method to minimize coefficients in forms

Classified possible divisor forms for quadratic representations

Abstract

In this article, Joseph-Louis Lagrange analyzed those numbers which may be represented by the quadratic form $Bt^2 + Ctu + Du^2$. After proving a few theorems on the divisors of such numbers (and their possible forms), Lagrange developed a method for minimizing $C$ in order to determine systematically the possible forms for the divisors of numbers represented by the forms $t^2 \pm au^2$. This article was originally published as "Recherches d'Arithm\'etique" in the 1773 volume of the Nouveaux M\'emoires de l'Acad\'emie royale des Sciences et Belles-Lettres de Berlin (printed 1775), pp. 265-312. It is the first of two articles on this subject, the latter of which appeared in the 1775 volume of the Nouveaux M\'emoires.