The Parents of Jacobi%26apos;s Four Squares Theorem Are Unique

作者：Williams Kenneth S^*

来源：American Mathematical Monthly, 2013, 120(4): 329-345.

DOI：10.4169/amer.math.monthly.120.04.329

Jacobi%26apos;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%26apos;s four squares formula can be deduced by equating coefficients of qn (the %26quot;parents%26quot;) 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%26apos;s four triangular numbers theorem. Finally, a general uniqueness result is proved.

• 出版日期2013-4