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Elementary Number Theory: Primes, Congruences, and Secrets: A Computational Approach

William Stein
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Allen Stenger
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This is a brief introductory textbook in number theory, that covers only a few topics but goes into depth on those. It includes a number of recent developments, such as RSA encryption and elliptic curve factorization.

The book is subtitled “A Computational Approach,” but the computational part is poorly integrated. If you quietly erased all the computer parts it would be very much the same book, although with fewer examples. The book has many examples of using the open-source mathematical software package Sage, but these seem intended more to demonstrate Sage than to illuminate number theory. From the subtitle I expected an “experimental mathematics” approach but that is absent. (A good experimental elementary number theory book is Burn’s A Pathway into Number Theory.)

There must be hundreds of elementary number theory books by now (MAA Reviews lists 61), and when a new one comes out you have to ask yourself, “What’s special about this one?” Davenport’s The Higher Arithmetic has been around forever and has similar coverage to this book, including a chapter on computers in the theory of numbers that covers RSA encryption and elliptic curve factorization. It’s hard to argue that the present book represents an advance in number theory textbooks.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.


Preface.- Prime Numbers.- The Ring of Integers Modulo n.- Public-Key Cryptography.- Quadratic Reciprocity.- Continued Fractions.- Elliptic Curves.- Answers and Hints.- References.