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Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic 2018 ed. [Kietas viršelis]

2.00/5 (2 ratings by Goodreads)
  • Formatas: Hardback, 279 pages, aukštis x plotis: 235x155 mm, weight: 736 g, 9 Illustrations, color; 15 Illustrations, black and white; XVIII, 279 p. 24 illus., 9 illus. in color., 1 Hardback
  • Serija: Undergraduate Texts in Mathematics
  • Išleidimo metai: 10-Dec-2018
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030026035
  • ISBN-13: 9783030026035
Kitos knygos pagal šią temą:
Pythagorean Introduction to Number Theory: Right Triangles, Sums of Squares, and Arithmetic 2018 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 279 pages, aukštis x plotis: 235x155 mm, weight: 736 g, 9 Illustrations, color; 15 Illustrations, black and white; XVIII, 279 p. 24 illus., 9 illus. in color., 1 Hardback
  • Serija: Undergraduate Texts in Mathematics
  • Išleidimo metai: 10-Dec-2018
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030026035
  • ISBN-13: 9783030026035
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030026035.html
Right triangles are at the heart of this textbooks 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 ofwhich 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 readers appreciation of the final chapters.

Recenzijos

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 eld 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 Crāmreanu, 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 substantial supply of examples. the book has several other potential uses: it could be used as a text for a second semester course in number theory or special topics course, or as a text for an introductory graduate course. Its also just an interesting book to have on ones shelf. (Mark Hunacek, MAA Reviews, June 24, 2019)

Part I Foundational material
1 Introduction
3(10)
1.1 The Pythagorean Theorem
3(3)
1.2 Pythagorean triples
6(2)
1.3 The questions
8(1)
Exercises
8(2)
Notes
10(3)
2 Basic number theory
13(46)
2.1 Natural numbers, mathematical induction, and the Well-ordering Principle
13(1)
2.2 Divisibility and prime factorization
14(8)
2.3 The Chinese Remainder Theorem
22(2)
2.4 Euler's Theorem
24(6)
2.5 Polynomials modulo a prime
30(2)
2.6 Digit expansions
32(7)
2.7 Digit expansions of rational numbers
39(2)
2.8 Primitive roots
41(8)
Exercises
49(4)
Notes
53(6)
3 Integral solutions to the Pythagorean Equation
59(22)
3.1 Solutions
59(2)
3.2 Geometric method to find solutions
61(4)
3.3 Geometric method to find solutions: Non-Pythagorean examples
65(5)
3.4 Application: X4 + Y4 = z4
70(2)
Exercises
72(1)
Notes
73(8)
4 What integers are areas of right triangles?
81(10)
4.1 Congruent numbers
81(2)
4.2 Small numbers
83(1)
4.3 Connection to cubic equations
84(3)
Exercises
87(1)
Notes
88(3)
5 What numbers are the edges of a right triangle?
91(14)
5.1 The theorem
91(2)
5.2 Gaussian integers
93(2)
5.3 The proof of Theorem 5.2
95(3)
5.4 Irreducible elements in Z[ i]
98(1)
5.5 Proof of Theorem 5.1
99(2)
Exercises
101(1)
Notes
102(3)
6 Primes of the form 4k + 1
105(14)
6.1 Euclid's theorem on the infinitude of primes
105(2)
6.2 Quadratic residues
107(5)
6.3 An application of the Law of Quadratic Reciprocity
112(1)
Exercises
113(2)
Notes
115(4)
7 Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol
119(14)
7.1 Gauss sums and Quadratic Reciprocity
119(5)
7.2 The Jacobi Symbol
124(5)
Exercises
129(1)
Notes
130(3)
Part II Advanced Topics
8 Counting Pythagorean triples modulo an integer
133(18)
8.1 The Pythagorean Equation modulo a prime number p
133(5)
8.2 Solutions modulo n for a natural number n
138(7)
Exercises
145(1)
Notes
146(5)
9 How many lattice points are there on a circle or a sphere?
151(14)
9.1 The case of two squares
151(4)
9.2 More than two squares
155(1)
9.3 Integral points on arcs
156(6)
Exercises
162(2)
Notes
164(1)
10 What about geometry?
165(22)
10.1 Lattices in Rn
165(3)
10.2 Minkow ski's Theorem
168(4)
10.3 Sums of two squares
172(1)
10.4 Sums of four squares
173(3)
10.5 Sums of three squares
176(4)
Exercises
180(2)
Notes
182(5)
11 Another proof of the four squares theorem
187(8)
11.1 Quatemions
187(2)
11.2 Matrix representation
189(1)
11.3 Four squares
190(2)
Exercises
192(1)
Notes
193(2)
12 Quadratic forms and sums of squares
195(16)
12.1 Quadratic forms with integral coefficients
195(5)
12.2 Binary forms
200(3)
12.3 Ternary forms
203(3)
12.4 Three squares
206(2)
Exercises
208(1)
Notes
209(2)
13 How many Pythagorean triples are there?
211(16)
13.1 The asymptotic formula
211(6)
13.2 The computation of C2
217(3)
Exercises
220(3)
Notes
223(4)
14 How are rational points distributed, really?
227(20)
14.1 The real line
227(13)
14.2 The unit circle
240(3)
Exercises
243(2)
Notes
245(2)
Appendix A: Background 247(8)
Appendix B: Algebraic integers 255(6)
Appendix C: SageMath 261(10)
References 271(6)
Index 277
Ramin Takloo-Bighash is a Professor of Mathematics at the University of Illinois at Chicago, where his research is centered on Diophantine geometry and automorphic forms. His enthusiasm for research inspires his teaching at all levels.