Sums Of Squares Of Integers

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Sums Of Squares Of Integers

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Author : Carlos J. Moreno
language : en
Publisher: CRC Press
Release Date : 2005-12-09

Sums Of Squares Of Integers written by Carlos J. Moreno and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-12-09 with Mathematics categories.

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.

Representations Of Integers As Sums Of Squares

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Author : E. Grosswald
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Representations Of Integers As Sums Of Squares written by E. Grosswald and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

From Polynomials To Sums Of Squares

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Author : T.H Jackson
language : en
Publisher: CRC Press
Release Date : 1995-01-01

From Polynomials To Sums Of Squares written by T.H Jackson and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-01-01 with Mathematics categories.

From Polynomials to Sums of Squares describes a journey through the foothills of algebra and number theory based around the central theme of factorization. The book begins by providing basic knowledge of rational polynomials, then gradually introduces other integral domains, and eventually arrives at sums of squares of integers. The text is complemented with illustrations that feature specific examples. Other than familiarity with complex numbers and some elementary number theory, very little mathematical prerequisites are needed. The accompanying disk enables readers to explore the subject further by removing the tedium of doing calculations by hand. Throughout the text there are practical activities involving the computer.

Concrete Mathematics

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Author : Ronald L. Graham
language : zh-CN
Publisher:
Release Date : 2020

Concrete Mathematics written by Ronald L. Graham and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Computer science categories.

A Pythagorean Introduction To Number Theory

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Author : Ramin Takloo-Bighash
language : en
Publisher: Springer
Release Date : 2018-11-26

A Pythagorean Introduction To Number Theory written by Ramin Takloo-Bighash and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-26 with Mathematics categories.

Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed. Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath. By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.

The Sensual Quadratic Form

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Author : John Horton Conway
language : en
Publisher: American Mathematical Soc.
Release Date : 1997-12-31

The Sensual Quadratic Form written by John Horton Conway and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-12-31 with Mathematics categories.

John Horton Conway's unique approach to quadratic forms was the subject of the Hedrick Lectures that he gave in August of 1991 at the Joint Meetings of the Mathematical Association of America and the American Mathematical Society in Orono, Maine. This book presents the substance of those lectures. The book should not be thought of as a serious textbook on the theory of quadratic forms. It consists rather of a number of essays on particular aspects of quadratic forms that have interested the author. The lectures are self-contained and will be accessible to the generally informed reader who has no particular background in quadratic form theory. The minor exceptions should not interrupt the flow of ideas. The afterthoughts to the lectures contain discussion of related matters that occasionally presuppose greater knowledge.

Discrete Mathematics And Its Applications

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Author : Kenneth Rosen
language : en
Publisher: McGraw-Hill Education
Release Date : 2006-07-26

Discrete Mathematics And Its Applications written by Kenneth Rosen and has been published by McGraw-Hill Education this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-07-26 with Mathematics categories.

Discrete Mathematics and its Applications, Sixth Edition, is intended for one- or two-term introductory discrete mathematics courses taken by students from a wide variety of majors, including computer science, mathematics, and engineering. This renowned best-selling text, which has been used at over 500 institutions around the world, gives a focused introduction to the primary themes in a discrete mathematics course and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications...from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields.

Solved And Unsolved Problems In Number Theory

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Author : Daniel Shanks
language : en
Publisher: American Mathematical Society
Release Date : 2024-01-24

Solved And Unsolved Problems In Number Theory written by Daniel Shanks and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-01-24 with Mathematics categories.

The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.

Elementary Number Theory With Applications

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Author : Thomas Koshy
language : en
Publisher: Academic Press
Release Date : 2002

Elementary Number Theory With Applications written by Thomas Koshy and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.

Elementary Number Theory focuses on number theory's role in the rapid development of art, coding theory, cryptology, computer science, and other necessities of modern life - confirming that human ingenuity and creativity are boundless.