Representations Of Integers As Sums Of Squares

Author : E. Grosswald
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 44,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461385660

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Representations of Integers as Sums of Squares by E. Grosswald Pdf

During the academic year 1980-1981 I was teaching at the Technion-the Israeli Institute of Technology-in Haifa. The audience was small, but con sisted of particularly gifted and eager listeners; unfortunately, their back ground varied widely. What could one offer such an audience, so as to do justice to all of them? I decided to discuss representations of natural integers as sums of squares, starting on the most elementary level, but with the inten tion of pushing ahead as far as possible in some of the different directions that offered themselves (quadratic forms, theory of genera, generalizations and modern developments, etc.), according to the interests of the audience. A few weeks after the start of the academic year I received a letter from Professor Gian-Carlo Rota, with the suggestion that I submit a manuscript for the Encyclopedia of Mathematical Sciences under his editorship. I answered that I did not have a ready manuscript to offer, but that I could use my notes on representations of integers by sums of squares as the basis for one. Indeed, about that time I had already started thinking about the possibility of such a book and had, in fact, quite precise ideas about the kind of book I wanted it to be.

Author : Carlos J. Moreno,Samuel S. Wagstaff Jr.
Publisher : CRC Press
Page : 363 pages
File Size : 51,5 Mb
Release : 2005-12-09
Category : Computers
ISBN : 9781420057232

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Sums of Squares of Integers by Carlos J. Moreno,Samuel S. Wagstaff Jr. Pdf

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 element

Author : Emil Grosswald
Publisher : Unknown
Page : 251 pages
File Size : 41,6 Mb
Release : 1985-01-01
Category : Forms, Quadratic
ISBN : 3540961267

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Representations of Integers as Sums of Squares by Emil Grosswald Pdf

Author : Richard Guy
Publisher : Springer Science & Business Media
Page : 455 pages
File Size : 53,8 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9780387266770

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Unsolved Problems in Number Theory by Richard Guy Pdf

Mathematics is kept alive by the appearance of new, unsolved problems. This book provides a steady supply of easily understood, if not easily solved, problems that can be considered in varying depths by mathematicians at all levels of mathematical maturity. This new edition features lists of references to OEIS, Neal Sloane’s Online Encyclopedia of Integer Sequences, at the end of several of the sections.

Author : Carlos J. Moreno,Samuel S. Wagstaff, Jr.
Publisher : CRC Press
Page : 368 pages
File Size : 40,7 Mb
Release : 2005-12-09
Category : Mathematics
ISBN : 9781584884569

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Sums of Squares of Integers by Carlos J. Moreno,Samuel S. Wagstaff, Jr. Pdf

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.

Author : George E. Andrews
Publisher : 清华大学出版社有限公司
Page : 684 pages
File Size : 45,8 Mb
Release : 2004
Category : Functions, Special
ISBN : 7302090890

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特殊函数 by George E. Andrews Pdf

Author : Janos Suranyi,Paul Erdös
Publisher : Springer Science & Business Media
Page : 322 pages
File Size : 51,5 Mb
Release : 2003-01-14
Category : Mathematics
ISBN : 0387953205

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Topics in the Theory of Numbers by Janos Suranyi,Paul Erdös Pdf

Number theory, the branch of mathematics that studies the properties of the integers, is a repository of interesting and quite varied problems, sometimes impossibly difficult ones. In this book, the authors have gathered together a collection of problems from various topics in number theory that they find beautiful, intriguing, and from a certain point of view instructive.

Author : Kenneth S. Williams
Publisher : Cambridge University Press
Page : 307 pages
File Size : 41,6 Mb
Release : 2011
Category : Mathematics
ISBN : 9781107002531

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Number Theory in the Spirit of Liouville by Kenneth S. Williams Pdf

A gentle introduction to Liouville's powerful method in elementary number theory. Suitable for advanced undergraduate and beginning graduate students.

Author : L J P Kilford
Publisher : World Scientific Publishing Company
Page : 252 pages
File Size : 43,7 Mb
Release : 2015-03-12
Category : Mathematics
ISBN : 9781783265473

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Modular Forms by L J P Kilford Pdf

Modular Forms is a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it. This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.

Author : David Burton
Publisher : McGraw Hill
Page : 453 pages
File Size : 41,7 Mb
Release : 2010-06-16
Category : Mathematics
ISBN : 9780077145088

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EBOOK: Elementary Number Theory by David Burton Pdf

Elementary Number Theory, Seventh Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.

Author : Władysław Narkiewicz
Publisher : Springer Science & Business Media
Page : 659 pages
File Size : 42,6 Mb
Release : 2011-09-02
Category : Mathematics
ISBN : 9780857295323

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Rational Number Theory in the 20th Century by Władysław Narkiewicz Pdf

The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat’s problem. Rational Number Theory in the 20th Century: From PNT to FLT offers a short survey of 20th century developments in classical number theory, documenting between the proof of the Prime Number Theorem and the proof of Fermat's Last Theorem. The focus lays upon the part of number theory that deals with properties of integers and rational numbers. Chapters are divided into five time periods, which are then further divided into subject areas. With the introduction of each new topic, developments are followed through to the present day. This book will appeal to graduate researchers and student in number theory, however the presentation of main results without technicalities will make this accessible to anyone with an interest in the area.

Author : Richard A. Mollin
Publisher : CRC Press
Page : 382 pages
File Size : 50,9 Mb
Release : 2008-02-21
Category : Mathematics
ISBN : 9781420066616

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Fundamental Number Theory with Applications by Richard A. Mollin Pdf

An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition • Removal of all advanced material to be even more accessible in scope • New fundamental material, including partition theory, generating functions, and combinatorial number theory • Expanded coverage of random number generation, Diophantine analysis, and additive number theory • More applications to cryptography, primality testing, and factoring • An appendix on the recently discovered unconditional deterministic polynomial-time algorithm for primality testing Taking a truly elementary approach to number theory, this text supplies the essential material for a first course on the subject. Placed in highlighted boxes to reduce distraction from the main text, nearly 70 biographies focus on major contributors to the field. The presentation of over 1,300 entries in the index maximizes cross-referencing so students can find data with ease.

Author : Ranjan Roy
Publisher : Cambridge University Press
Page : 491 pages
File Size : 45,9 Mb
Release : 2017-04-18
Category : Mathematics
ISBN : 9781107159389

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Elliptic and Modular Functions from Gauss to Dedekind to Hecke by Ranjan Roy Pdf

A thorough guide to elliptic functions and modular forms that demonstrates the relevance and usefulness of historical sources.

Author : Bruce C. Berndt,Ken Ono
Publisher : American Mathematical Soc.
Page : 290 pages
File Size : 40,9 Mb
Release : 2001
Category : Mathematics
ISBN : 9780821827468

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$q$-Series with Applications to Combinatorics, Number Theory, and Physics by Bruce C. Berndt,Ken Ono Pdf

The subject of $q$-series can be said to begin with Euler and his pentagonal number theorem. In fact, $q$-series are sometimes called Eulerian series. Contributions were made by Gauss, Jacobi, and Cauchy, but the first attempt at a systematic development, especially from the point of view of studying series with the products in the summands, was made by E. Heine in 1847. In the latter part of the nineteenth and in the early part of the twentieth centuries, two Englishmathematicians, L. J. Rogers and F. H. Jackson, made fundamental contributions. In 1940, G. H. Hardy described what we now call Ramanujan's famous $ 1\psi 1$ summation theorem as ``a remarkable formula with many parameters.'' This is now one of the fundamental theorems of the subject. Despite humble beginnings,the subject of $q$-series has flourished in the past three decades, particularly with its applications to combinatorics, number theory, and physics. During the year 2000, the University of Illinois embraced The Millennial Year in Number Theory. One of the events that year was the conference $q$-Series with Applications to Combinatorics, Number Theory, and Physics. This event gathered mathematicians from the world over to lecture and discuss their research. This volume presents nineteen of thepapers presented at the conference. The excellent lectures that are included chart pathways into the future and survey the numerous applications of $q$-series to combinatorics, number theory, and physics.

Author : G. H. Hardy,E. M. Wright,Joseph Silverman
Publisher : Oxford University Press
Page : 645 pages
File Size : 42,8 Mb
Release : 2008-07-31
Category : Mathematics
ISBN : 9780199219865

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An Introduction to the Theory of Numbers by G. H. Hardy,E. M. Wright,Joseph Silverman Pdf

The sixth edition of the classic undergraduate text in elementary number theory includes a new chapter on elliptic curves and their role in the proof of Fermat's Last Theorem, a foreword by Andrew Wiles and extensively revised and updated end-of-chapter notes.