Research on Arithmetic

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.