© 2018

A Pythagorean Introduction to Number Theory

Right Triangles, Sums of Squares, and Arithmetic


  • 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


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

Table of contents

  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
  3. Advanced Topics

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

About this book


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.


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

About the authors

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
  • 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 Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


“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 Crâşmăreanu, 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. It’s also just an interesting book to have on one’s shelf.” (Mark Hunacek, MAA Reviews, June 24, 2019)