Number Theory pp 541-586

Connections with Number Theory

• W. A. Coppel
Chapter
Part of the Universitext book series (UTX)

Abstract

In Proposition II.40 we proved Lagrange’s theorem that every positive integer can be represented as a sum of 4 squares. Jacobi (1829), at the end of his Fundamenta Nova, gave a completely different proof of this theorem with the aid of theta functions. Moreover, his proof provided a formula for the number of different representations. Hurwitz (1896), by developing further the arithmetic of quaternions which was used in Chapter II, also derived this formula. Here we give Jacobi’s argument preference since, although it is less elementary, it is more powerful.

Keywords

Modular Form Elliptic Curve Rational Point Elliptic Curf Theta Function
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