# A Pythagorean Introduction to Number Theory

## Benefits

• Offers an innovative approach to elementary number theory motivated by right triangles

• Inspires students to explore number theory through investigation of concrete examples

• Provides historical context throughout, showing how ideas developed in the field

• Features numerous engaging exercises, including many designed for SageMath

Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xviii
2. ### Foundational material

1. Front Matter
Pages 1-1
2. Ramin Takloo-Bighash
Pages 3-12
3. Ramin Takloo-Bighash
Pages 13-58
4. Ramin Takloo-Bighash
Pages 59-79
5. Ramin Takloo-Bighash
Pages 81-89
6. Ramin Takloo-Bighash
Pages 91-104
7. Ramin Takloo-Bighash
Pages 105-118
8. Ramin Takloo-Bighash
Pages 119-130

1. Front Matter
Pages 131-131
2. Ramin Takloo-Bighash
Pages 133-149
3. Ramin Takloo-Bighash
Pages 151-164
4. Ramin Takloo-Bighash
Pages 165-185
5. Ramin Takloo-Bighash
Pages 187-194
6. Ramin Takloo-Bighash
Pages 195-210
7. Ramin Takloo-Bighash
Pages 211-226
8. Ramin Takloo-Bighash
Pages 227-246
4. Back Matter
Pages 247-279

### Introduction

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.

### Keywords

Pythagorean theorem Elementary number theory Fermat's theorem Diophantine equation Primes of the form 4k+1 Quadratic reciprocity Pythagorean triples Counting lattice points Gauss sums Four squares theorem Quadratic forms and sums of squares Number theory with SageMath

#### Authors and affiliations

1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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.

### Bibliographic information

• Book Title A Pythagorean Introduction to Number Theory
• Book Subtitle Right Triangles, Sums of Squares, and Arithmetic
• Authors Ramin Takloo-Bighash
• Series Title Undergraduate Texts in Mathematics
• Series Abbreviated Title Undergraduate Texts Mathematics
• DOI https://doi.org/10.1007/978-3-030-02604-2
• Copyright Information Springer Nature Switzerland AG 2018
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Hardcover ISBN 978-3-030-02603-5
• Softcover ISBN 978-3-030-80529-6
• eBook ISBN 978-3-030-02604-2
• Series ISSN 0172-6056
• Series E-ISSN 2197-5604
• Edition Number 1
• Number of Pages XVIII, 279
• Number of Illustrations 15 b/w illustrations, 9 illustrations in colour
• Topics
• Buy this book on publisher's site
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