The Parents of Jacobi's Four Squares Theorem Are Unique

Abstract

Jacobi's four squares theorem asserts that the number of representations of a positive integer n as a sum of four squares is 8 times the sum of the positive divisors of n, which are not multiples of 4. A formula expressing an infinite product as an infinite sum is called a product-to-sum identity. The product-to-sum identities in a single complex variable q from which Jacobi's four squares formula can be deduced by equating coefficients of qⁿ (the “parents”) are explored using some amazing identities of Ramanujan, and are shown to be unique in a certain sense, thereby justifying the title of this article. The same is done for Legendre's four triangular numbers theorem. Finally, a general uniqueness result is proved.

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Kenneth S. Williams

KENNETH S. WILLIAMS received his Ph.D. degree in mathematics from the University of Toronto in 1965. In 1979 he was awarded his D.Sc. degree from the University of Birmingham, England for his research in the theory of numbers. He retired in 2002 after 36 years as a faculty member at Carleton University and is currently Professor Emeritus and Distinguished Research Professor. His interests include birding with his wife, gardening, and, once a week, looking after his three year old granddaughter Isabelle Sofie Olsen (with his wife's help of course).