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Recent Progress in the Study of Representations of Integers as Sums of Squares
Author(s) -
Chan Heng Huat,
Krattenthaler Christian
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609305004820
Subject(s) - mathematics , sketch , modular design , mathematics subject classification , subject (documents) , algebra over a field , combinatorics , pure mathematics , algorithm , computer science , operating system , library science
In this article, the authors collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. They start with a sketch of Milne's proof of two of these conjectures, and they also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of, respectively, 4 s 2 or 4 s ( s +1) triangular numbers. This approach is inspired by Zagier's proof of the Kac–Wakimoto formulas via modular forms. The survey ends with recent conjectures of the first author and Chua. 2000 Mathematics Subject Classification 11E25, 11F11.