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A Pythagorean Introduction to Number Theory

A Pythagorean Introduction to Number Theory

Wydawnictwo Springer Nature Customer Service Center GmbH
Data wydania 01/12/2018
Wydanie Pierwsze
Liczba stron 279
Forma publikacji książka w twardej oprawie
Poziom zaawansowania Do szkoleń zawodowych, hobbystycznych i naukowych
Język angielski
ISBN 9783030026035
Kategorie Teoria liczb
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Opis książki

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 book reads well. ... I would think this would be an enjoyable book from which to teach, since it covers the standard material in the way in which mathematics is done by asking questions and then developing the theory necessary for answering the questions. ... All in all, this is a very good antidote to the definition-theorem-proof approach to introductions to various subdisciplines of mathematics." (Duncan A. Buell, Mathematical Reviews, February, 2020) "The present book has a high didactic quality being a detailed proof that number theory is 'a field of study that sits at the crossroads of many branches of mathematics, and that fact makes a prominent appearance in this book' as is pointed out in Preface. ... this book develops step by step a large number of techniques to solve a lot of number theoretic problems. ... this book is an excellent source for all readers interested in number theory." (Mircea Crasmareanu, zbMath 1410.11002, 2019) "This book offers an interesting variation on the traditional undergraduate number theory course. ... The book is quite nicely written, with good motivation and a subst

A Pythagorean Introduction to Number Theory

Spis treści

Part I Foundational Material.- 1. Introduction.- 2. Basic number theory.- 3. Integral solutions to the Pythagorean Equation.- 4. What integers are areas of right triangles?.- 5. What numbers are the edges of a right triangle?.- 6. Primes of the form 4k+1.- 7. Gauss sums, Quadratic Reciprocity, and the Jacobi symbol.- Part II Advanced Topics.- 8. Counting Pythagorean triples modulo an integer.- 9. How many lattice points are there on a circle or a sphere?.- 10. What about geometry?.- 11. Another proof of the four squares theorem.- 12. Quadratic forms and sums of squares.- 13. How many Pythagorean triples are there?.- 14. How are rational points distributed, really?.- Part III Appendices.- A. Background.- B. Algebraic integers.- C. SageMath.- References.- Index.

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