An Overview on Italian Arithmetic after the Disquisitiones Arithmeticae

Abstract

Thedecades around 1800were not a period inwhich puremathematics in general, and number theory in particular, flourished in Italy, see [Bottazzini 1994]. It is significant in this respect that Joseph Louis Lagrange, whose birth and early studies took place in Torino, finally became a prominent representative of the Frenchmathematical school and that, decades later, Guglielmo Libri still spent most of his academic career in France. Thus, Gauss’s Disquisitiones Arithmeticae did not have an immediate resonance in Italian mathematical circles. Gianfrancesco Malfatti, a professor in Ferrara, already seventy years old at the time of the publication of theDisquisitiones Arithmeticae, was one of the rare Italian mathematicians who manifested some interest in arithmetic, and above all in the theory of algebraic equations, in the last decades of the eighteenth century. In 1804, he criticized Ruffini’s first attempts to prove that equations of degree higher than 4 are not solvable by radicals and thus contributed to Ruffini’s later works. A year later, Malfatti published what was probably the best Italian paper on arithmetical matters at that time, [Malfatti 1805], in which he studied the following problem using simple divisibility properties: multiply a given integer by a square, in such a way that it may be expressed as a sum of a given number of squares. However, none of these papers mentions Gauss and his masterwork. In fact, Malfatti did not display a serious knowledge of the recent developments in number theory; the short historical discussion in his 1805 paper only says: