A Pythagorean Introduction to Number Theory

Right Triangles, Sums of Squares, and Arithmetic

  • Ramin Takloo-Bighash

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

  • Ramin Takloo-Bighash
    • 1
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

Bibliographic information

  • DOI
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-02603-5
  • Online ISBN 978-3-030-02604-2
  • Series Print ISSN 0172-6056
  • Series Online ISSN 2197-5604
  • Buy this book on publisher's site