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Sums of Squares of Integers covers topics in combinatorial amount precept as they relate to counting representations of integers as sums of a positive amount of squares. The book introduces a stimulating area of amount precept the place evaluation continues to proliferate. It is a book of "firsts" – particularly it is the first book to combine Liouville's elementary methods with the analytic methods of modular options to evaluate the illustration of integers as sums of squares. It is the primary book to tell strategies to compute the amount of representations of an integer n as a result of the sum of s squares of integers for any s and n. Moreover it is the first book to offer a proof of Szemeredi's theorem, and is the first amount precept book to debate how the trendy precept of modular varieties enhances and clarifies the classical elementary outcomes about sums of squares.
The book presents a quantity of current, however nonetheless fascinating and instructive, examples of modular varieties. Two chapters develop useful properties of the Bernoulli numbers and illustrate arithmetic progressions, proving the theorems of van der Waerden, Roth, and Szemeredi. The book moreover explains functions of the thought to three points that lie outdoor of amount idea inside the areas of cryptanalysis, microwave radiation, and diamond chopping. The textual content material is complemented by the inclusion of over one hundred exercises to examine the reader's understanding.
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