Talteori

This Book of Number Theory is a captivating exploration of one of the oldest branches of mathematics. From its ancient origins to modern-day breakthroughs, this book uncovers the patterns, properties, and relationships that lie at the heart of numbers. It delves into topics such as divisibility, prime numbers, modular arithmetic, Diophantine equations, prime number distribution, and sieve methods. With its comprehensive coverage and engaging explanations, The Book of Number Theory reveals the beauty and significance of this fascinating mathematical discipline.

Dieses Buch behandelt die Grundlagen der Algebra und der elementaren Zahlentheorie. Zentrale Begriffe sind Primelemente und irreduzible Elemente. Ausgehend vom Aufbau einer Arithmetik in Hauptidealringen und insbesondere euklidischen Ringen sind die zentralen Themen zum einen irreduzible Polynome, zum anderen Primzahlen. Dies führt zu den algebraischen Körpererweiterungen und zu Fragen nach der Konstruierbarkeit mit Zirkel und Lineal. Nach einem längeren Ausflug in die Gruppentheorie bis zum Sylow-Satz und den auflösbaren Gruppen wird die Idee der Galoistheorie exemplarisch an der Frage der Auflösbarkeit von Polynomgleichungen behandelt. Zentrale Themen der Zahlentheorie sind Verteilung und Eigenschaften von Primzahlen, Primzahltests, Kongruenzen, der Chinesische Restsatz, quadratische Reste bis hin zum quadratischen Reziprozitätsgesetz und Lösungsansätze für einige diophantische Gleichungen. Neu in dieser 3. Auflage ist neben einer Erweiterung des Themas "Konstruktion mit Zirkel und Lineal", dass es begleitend zum Buch ein Audio-Angebot gibt. Die Audiodateien können im E-Book direkt durch Anklicken angehört werden, für das gedruckte Buch steht die SN More Media App zur Verfügung.

This book was originally published in 2006. Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory. The theory has evolved to describe the relationship between finite groups, modular forms and vertex operator algebras. Moonshine Beyond the Monster describes the general theory of Moonshine and its underlying concepts, emphasising the interconnections between mathematics and mathematical physics. Written in a clear and pedagogical style, this book is ideal for graduate students and researchers working in areas such as conformal field theory, string theory, algebra, number theory, geometry and functional analysis. Containing over a hundred exercises, it is also a suitable textbook for graduate courses on Moonshine and as supplementary reading for courses on conformal field theory and string theory.

There is no surprise that arithmetic properties of integral ("whole") numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other "better known" numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of ¿ and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible - almost popular - level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.

A prime number is any integer greater than one which has only itself and one as factors. For example, 5 is prime since it does not have 2, 3, or 4 as a factor. 1 and 5 are the only factors of 5.Prime numbers have been studied since antiquity. Eratosthenes of Cyrene lived about 276 BCE to 194 BCE. He developed a method of finding prime numbers that is still taught today. It is called the Sieve of Eratosthenes.Prime numbers continue to fascinate mathematicians today. Prime numbers are used in number theory, and in cryptography.

This volume presents the revised papers of the 14th International Conference in Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, MCQMC 2020, which took place online during August 10-14, 2020. This book is an excellent reference resource for theoreticians and practitioners interested in solving high-dimensional computational problems, arising, in particular, in statistics, machine learning, finance, and computer graphics, offering information on the latest developments in Monte Carlo and quasi-Monte Carlo methods and their randomized versions.

This book is an autobiographical interview with Chinese Academician Yuan Wang on his experience in mathematical research. The book looks back on Wang's collaboration with his teacher Hua Loo-Keng and younger scholars, offering insights into fruitful cooperation in mathematical research.In this book, Yuan Wang¿s path of studying Goldbach conjecture is revealed in detail from motivation to method. Then his work on algebraic number theory is traced back in a separate chapter. The book ends with two chapters which introduce Wang¿s interest in history of mathematics and his hobbies outside of mathematical research. Wang shows how a mathematician can in the same time be a historical and popular science writer and, in particular, a well-received calligrapher. The book is intended for undergraduate and graduate students studying number theory. Researchers who are willing to learn from the experience of an established mathematician, as well as math amateurs and general audience who are interested in Yuan Wang's life story might also find this book appealing.

In recent years, extensive research has been conducted by eminent mathematicians and engineers whose results and proposed problems are presented in this new volume. It is addressed to graduate students, research mathematicians, physicists, and engineers. Individual contributions are devoted to topics of approximation theory, functional equations and inequalities, fixed point theory, numerical analysis, theory of wavelets, convex analysis, topology, operator theory, differential operators, fractional integral operators, integro-differential equations, ternary algebras, super and hyper relators, variational analysis, discrete mathematics, cryptography, and a variety of applications in interdisciplinary topics. Several of these domains have a strong connection with both theories and problems of linear and nonlinear optimization. The combination of results from various domains provides the reader with a solid, state-of-the-art interdisciplinary reference to theory and problems.  Some of the works provide guidelines for further research and proposals for new directions and open problems with relevant discussions.

"Computing with Fermat" is a fascinating collection of math articles that pay tribute to the great Pierre de Fermat, exploring various math problems using computational number theory that relate to his work. With chapters such as "On Fermat's Factorization Method," "Fun With the Sqrt(n) Primality Test," "Near-Misses of Fermat's Last Theorem," and "Marin Mersenne and the Power of Modern Computing," this book offers a comprehensive look at how computational number theory is changing the game in the world of mathematics.The author, who found solace in Simon Singh's book "Fermat's Enigma" during a difficult time in his life, was inspired to delve into the world of computational number theory. The book features the author's own papers related to Fermat's work, as well as two chapters from his book "The Lowbrow Experimental Mathematician" that explore certain Fermat-type problems in elementary number theory."Computing with Fermat" is a must-read for anyone interested in the legacy of Pierre de Fermat and the fascinating world of computational number theory. With its accessible writing style and engaging content, this book is sure to inspire and entertain readers of all levels.

In the two-volume set 'A Selection of Highlights' we present basics of mathematics in an exciting and pedagogically sound way. This volume examines fundamental results in Algebra and Number Theory along with their proofs and their history. In the second edition, we include additional material on perfect and triangular numbers. We also added new sections on elementary Group Theory, p-adic numbers, and Galois Theory. A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications. Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested.

This proceedings volume contains articles related to the research presented at the 2019 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning non-abelian aspects This volume contains both original research articles as well as articles that contain both new research as well as survey some of these recent developments.

This book contains selected chapters on perfectoid spaces, their introduction and applications, as invented by Peter Scholze in his Fields Medal winning work. These contributions are presented at the conference on "Perfectoid Spaces" held at the International Centre for Theoretical Sciences, Bengaluru, India, from 9-20 September 2019. The objective of the book is to give an advanced introduction to Scholze's theory and understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, lifting of modular forms, completed cohomology, local Langlands program, and special values of L-functions. All chapters are contributed by experts in the area of arithmetic geometry that will facilitate future research in the direction.

This volume features contributions from the Women in Commutative Algebra (WICA) workshop held at the Banff International Research Station (BIRS) from October 20-25, 2019, run by the Pacific Institute of Mathematical Sciences (PIMS). The purpose of this meeting was for groups of mathematicians to work on joint research projects in the mathematical field of Commutative Algebra and continue these projects together long-distance after its close. The chapters include both direct results and surveys, with contributions from research groups and individual authors.The WICA conference was the first of its kind in the large and vibrant area of Commutative Algebra, and this volume is intended to showcase its important results and to encourage further collaboration among marginalized practitioners in the field. It will be of interest to a wide range of researchers, from PhD students to senior experts. 

"Searching for N" is a captivating collection of articles that takes readers on a journey through the fascinating world of number theory and beyond. Written by a mathematics enthusiast, this book presents classic topics like prime numbers and divisibility, but also includes surprises like short stories, anecdotes, and informational articles on a variety of different subjects.Readers will find a diverse range of chapters that explore intriguing topics such as Concrete Calculator-Word Primes, Summation of Infinite Series, Zen Master Bodhidharma, Iteration Problems, Barbelo and Gnosticism, Artificial Intelligence, and more. Each chapter attempts to offer a unique perspective on number theory, and other subjects.By delving into apparently unrelated subjects, the author attempts to uncover the interconnectedness of mathematics with our surroundings. Whether you are a seasoned mathematician or simply curious about numbers, this book is sure to engage your mind and spark your imagination.But "Searching for N" is more than just a book on number theory. It is also a story of perseverance and passion for mathematics. The author, Fitzroy, struggled with the complexity of number theory and found himself unable to solve the most challenging problems. However, he refused to give up on his dream of contributing something to the world of mathematics.Determined to find his own path, Fitzroy turned to writing as a way to express his ideas and insights. He began crafting short stories that explored the mysteries of the human psyche, often incorporating abstract concepts and philosophical ideas into his writing. Despite his struggles with mathematical problem-solving, a few of his stories resonated with readers and were admired for their emotional power and insight.Join Fitzroy on his journey through "Searching for N" and discover the beauty and power of number theory, and the importance of perseverance and passion in pursuing one's dreams.