Sum of Squa of Integers
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Sum of Squa of Integers

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Sums of Squares of Integers covers topics in combinatorial number theory as they relate to counting representations of integers as sums of a certain number of squares. The book introduces a stimulating area of number theory where research continues to proliferate. It is a book of "firsts" - namely it is the first book to combine Liouville's elementary methods with the analytic methods of modular functions to study the representation of integers as sums of squares. It is the first book to tell how to compute the number of representations of an integer n as the sum of s squares of integers for any s and n. It is also the first book to give a proof of Szemeredi's theorem, and is the first number theory book to discuss how the modern theory of modular forms complements and clarifies the classical fundamental results about sums of squares. The book presents several existing, yet still interesting and instructive, examples of modular forms. 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 also explains applications of the theory to three problems that lie outside of number theory in the areas of cryptanalysis, microwave radiation, and diamond cutting. The text is complemented by the inclusion of over one hundred exercises to test the reader's understanding.show more

Product details

  • Hardback | 368 pages
  • 152.4 x 236.2 x 25.4mm | 612.36g
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 7 black & white illustrations
  • 1584884568
  • 9781584884569

Review quote

"provides an excellent supplementary source of information for the reader, not least in the many biographical footnotes on the mathematicians involved in the subject matter, and there are also more than a thousand exercises and examples in the text. Besides various tables for results in computational number theory, the nine appendices cover material from set theory, which includes discussions on the axiom of choice and Zorn's Lemma, to the ABC conjecture." --P. Shiu, Zentralblatt MATH, Vol. 943 ides an excellent supplementary source of information for the reader, not least in the many biographical footnotes on the mathematicians involved in the subject matter, and there are also more than a thousand exercises and examples in the text. Besides various tables for results in computational number theory, the nine appendices cover material from set theory, which includes discussions on the axiom of choice and Zorn's Lemma, to the ABC conjecture." --P. Shiu, Zentralblatt MATH, Vol. 943 "This is a remarkable book that will be a valuable reference for many people, including me. The book shows great care in preparation, and the ample details and motivation will be appreciated by lots of students. The solid punches at the end of each chapter will be appreciated by everybody. It deserves success with many adoptions as a text." --Irving Kaplansky, Mathematical Sciences Research Institute at Berkeley is a remarkable book that will be a valuable reference for many people, including me. The book shows great care in preparation, and the ample details and motivation will be appreciated by lots of students. The solid punches at the end of each chapter will be appreciated by everybody. It deserves successwith many adoptions as a text." --Irving Kaplansky, Mathematical Sciences Research Institute at Berkeley "This is a great book! It can be used in many ways: for a university course at one extreme, and as selective light reading for pleasure at the other. The author's enthusiasm carries the reader along clearly and easily, spilling over to scores of fascinating, beautifully written footnotes, which include more than fifty mini-biographies. excellent and highly recommended." --Short Book Reviews, Vol. 21, No. 2, August, 2001 is a great book! It can be used in many ways: for a university course at one extreme, and as selective light reading for pleasure at the other. The author's enthusiasm carries the reader along clearly and easily, spilling over to scores of fascinating, beautifully written footnotes, which include more than fifty mini-biographies. excellent and highly recommended." --Short Book Reviews, Vol. 21, No. 2, August, 2001show more

Table of contents

Introduction Prerequisites Outline of Chapters 2 - 8 Elementary Methods Introduction Some Lemmas Two Fundamental Identities Euler's Recurrence for Sigma(n) More Identities Sums of Two Squares Sums of Four Squares Still More Identities Sums of Three Squares An Alternate Method Sums of Polygonal Numbers Exercises Bernoulli Numbers Overview Definition of the Bernoulli Numbers The Euler-MacLaurin Sum Formula The Riemann Zeta Function Signs of Bernoulli Numbers Alternate The von Staudt-Clausen Theorem Congruences of Voronoi and Kummer Irregular Primes Fractional Parts of Bernoulli Numbers Exercises Examples of Modular Forms Introduction An Example of Jacobi and Smith An Example of Ramanujan and Mordell An Example of Wilton: t (n) Modulo 23 An Example of Hamburger Exercises Hecke's Theory of Modular Forms Introduction Modular Group ? and its Subgroup ? 0 (N) Fundamental Domains For ? and ? 0 (N) Integral Modular Forms Modular Forms of Type Mk(? 0(N);chi) and Euler-Poincare series Hecke Operators Dirichlet Series and Their Functional Equation The Petersson Inner Product The Method of Poincare Series Fourier Coefficients of Poincare Series A Classical Bound for the Ramanujan t function The Eichler-Selberg Trace Formula l-adic Representations and the Ramanujan Conjecture Exercises Representation of Numbers as Sums of Squares Introduction The Circle Method and Poincare Series Explicit Formulas for the Singular Series The Singular Series Exact Formulas for the Number of Representations Examples: Quadratic Forms and Sums of Squares Liouville's Methods and Elliptic Modular Forms Exercises Arithmetic Progressions Introduction Van der Waerden's Theorem Roth's Theorem t 3 = 0 Szemeredi's Proof of Roth's Theorem Bipartite Graphs Configurations More Definitions The Choice of tm Well-Saturated K-tuples Szemeredi's Theorem Arithmetic Progressions of Squares Exercises Applications Factoring Integers Computing Sums of Two Squares Computing Sums of Three Squares Computing Sums of Four Squares Computing rs(n) Resonant Cavities Diamond Cutting Cryptanalysis of a Signature Scheme Exercises References Indexshow more

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