A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares

Abstract

Diophantus probably knew, and Lagrange [L] proved, that every positive integer can be written as a sum of four perfect squares. Jacobi [J] proved the stronger result that the number of ways in which a positive integer can be so written1 equals 8 times the sum of its divisors that are not multiples of 4. Here we give a short new proof that only uses high school algebra, and is completely from scratch. All infinite series and products that appear are to be taken in the entirely elementary sense of formal power series. The problem of representing integers as sums of squares has drawn the attention of many great mathematicians, and we encourage the reader to look up Grosswald's [G] erudite masterpiece on this subject. The crucial part of our proof is played by two simple identities, that we state as one Lemma.