This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergruate 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 infinitely many prime numbers. At the same time, he also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over 1000 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 1000 years later (in 1976), Diffie and Hellman introduced the first ever public-key cryptosystem, which enabled two people to communicate secretly over a public communications channel with no predetermined secret; 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, public-key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem.
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.
Title Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach
Author(s) William Stein
Publisher: Springer 2009 edition (December 3, 2008); eBook (2011)
Hardcover/Paperback 168 pages
eBook PDF (172 pages, 2.6 MB)
Language: English
ISBN-10: 0387855246
ISBN-13: 978-0387855240
Download: http://modular.math.washington.edu/ent/
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