Number Theory Through the Eyes of Sophie Germain

An Inquiry Course

David Pengelley

• Publisher: American Mathematical Society
• Year: 2023
• Print ISBN: 9781470472207
• eBook ISBN: 9781470473440
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Number Theory Through the Eyes of Sophie Germain: An Inquiry Course is an innovative textbook for an introductory number theory course.Sophie Germain (1776–1831) was largely self-taught in mathematics and, two centuries ago, in solitude, devised and implemented a plan to prove Fermat's Last Theorem. We have only recently completely understood this work from her unpublished letters and manuscripts. David Pengelley has been a driving force in unraveling this mystery and here he masterfully guides his readers along a path of discovery. Germain, because of her circumstances as the first woman to do important original mathematical research, was forced to learn most of what we now include in an undergraduate number theory course for herself. Pengelley has taken excerpts of her writings (and those of others) and, by asking his readers to decipher them, skillfully leads us through an inquiry-based course in elementary number theory. It is a detective story on multiple levels. What is Sophie Germain thinking? What do her mathematical writings mean? How do we understand what she knew and what she was trying to do, where she succeeded and where she didn't?Number Theory Through the Eyes of Sophie Germainis simultaneously a masterpiece of historical scholarship, a guide to reading and teaching from primary-source historical documents, an inquiry-based textbook for introductory number theory, and the riveting story of a major, but still unappreciated, mathematician. Work is required of the reader. Readers are carefully guided to discover and prove almost all results for themselves in a sequence of scaffolded exploratory tasks with hints, fully integrated with the narrative. The difficulty of the inquiry tasks varies considerably, but the author provides the reader with appropriately helpful guidance at every step. An introductory number theory course taught with this text would be a remarkable, potentially life-changing, experience.—Stephen Kennedy, Carleton College and MAA Press

Table of contents

• Cover
• 1/2 title page
• Title page
• Copyright page
• Dedication
• Contents
• Preface
• Chapter 1. Introduction
• Chapter 2. Sophie Germain, Number Theory, and Fermat’s Last Theorem
• Chapter 3. Germain’s Plan to Prove Fermat’s Last Theorem
• Chapter 4. Fermat’s Last Theorem for Exponent Four
• Chapter 5. Germain’s Grand Plan and a Letter to Gauss
• Chapter 6. Congruence, Germain’s Basic Lemma, Systems of Linear Congruences, and Higher Power Congruences
• Chapter 7. Primitive Roots
• Chapter 8. Germain Carrying out Her Grand Plan
• Chapter 9. Large Size of Solutions and Sophie Germain’s Theorem
• Chapter 10. The Demise of the Grand Plan: A Letter to Legendre
• Chapter 11. *Prime Patterns in Quadratic Forms
• Appendix A. *How Fermat Discovered his Theorem, and Other Divisibility Delights
• References
• Credits
• Index
• Back Cover