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Elementary Number Theory: Primes, Congruences, and Secrets

A Computational Approach

  • Textbook
  • © 2009

Overview

Authors:
  1. William Stein

    1. Dept. Mathematics, University of Washington, Seattle, U.S.A.

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

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Table of contents (6 chapters)

  1. Front Matter

    Pages 1-9
    Download chapter PDF

  2. Prime Numbers

    • William Stein
    Pages 1-20

  3. The Ring of Integers Modulo n

    • William Stein
    Pages 1-27

  4. Public-key Cryptography

    • William Stein
    Pages 1-20

  5. Quadratic Reciprocity

    • William Stein
    Pages 1-23

  6. Continued Fractions

    • William Stein
    Pages 1-30

  7. Elliptic Curves

    • William Stein
    Pages 1-25

  8. Back Matter

    Pages 1-18
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Keywords

About this book

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predeterminedsecret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles’ resolution of Fermat’s Last Theorem.

Reviews

From the reviews:

"This one treats topics that have become standard in recent years … and it has exercises with selected solutions. … it gives the students a tool to do calculations that illustrate even the most abstract concepts, and, simultaneously, introduces them to an open source software that can later be applied profitably for studying research problems. … introducing the reader to a powerful software system." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1155, 2009)

"The cliché that number theory, ever the purest mathematics, now yields very practical applications barely tells the story. Teach undergraduate number theory today, and students demand to hear about public-key cryptography and related technologies. … Stein (Univ. of Washington) serves undergraduates well by … opening the way by intimating their power. … he frames the sophisticated Birch and Swinnerton-Dyer conjecture as the new canonical challenge for the future. Summing Up: Recommended. All undergraduates students, professionals, and general readers." (D. V. Feldman, Choice, Vol. 47 (2), October, 2009)

"This book is an introduction to elementary number theory with a computational flavor. … Many numerical examples are given throughout the book using the Sage mathematical software. The text is aimed at an undergraduate student with a basic knowledge of groups, rings and fields. Each chapter concludes with several exercises." (Samuel S. Wagstaff Jr., Mathematical Reviews, Issue 2009 i)

Authors and Affiliations

  • Dept. Mathematics, University of Washington, Seattle, U.S.A.

    William Stein

Bibliographic Information

  • Book Title: Elementary Number Theory: Primes, Congruences, and Secrets

  • Book Subtitle: A Computational Approach

  • Authors: William Stein

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/b13279

  • Publisher: Springer New York, NY

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag New York 2009

  • Hardcover ISBN: 978-0-387-85524-0Published: 03 December 2008

  • Softcover ISBN: 978-1-4419-2752-1Published: 19 November 2010

  • eBook ISBN: 978-0-387-85525-7Published: 28 October 2008

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: X, 168

  • Number of Illustrations: 45 b/w illustrations

  • Topics: Number Theory, Algebraic Geometry

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