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Excursions in Number Theory

Dover Publications
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This is an introductory book in number theory that was published in 1966 as a popular math book, but in modern terms is somewhere between popular math and an introductory course. It assumes the reader is competent at high-school algebra (mostly polynomial equations) and radicals, but does not have any exercises and is not systematic. Despite being fifty years old, it still covers the mainstream of number theory. The present volume is a Dover 1988 corrected reprint of the 1966 Oxford edition.

The book covers most of the things you would expect in an elementary number theory book, such as divisibility, congruences, Pythagorean triples, Fermat’s little theorem, and Fermat’s Last Theorem (proved for exponent 4), along with some more unusual topics such as continued fractions. It also covers a few areas that are not really number theory, such as mental calculating prodigies and number curiosities (particular integers with unusual or unique properties). It omits a few recent developments you would expect to see in a current book, such as RSA encryption. The one aspect that is dated is the references to computers; these were just starting to be used in number theory when the book came out, but the subject has advanced tremendously since then and nearly all the computer facts are obsolete.

The book is divided into eleven chapters, each on a specific subject, but otherwise does not have much structure. Each chapter is a continuous narrative of interesting and related facts that may or may not be proved here. There’s also a 20-page notes section at the end that gives additional more advanced information about the topics discussed.

“They don’t make ‘em like this any more.” Authors of popular math books no longer assume their readers remember high-school algebra, and I don’t know any modern number theory books with a style similar to this one. There are a number of textbooks with similar coverage and prerequisites, such as Forman & Rash’s recent The Whole Truth About Whole Numbers, but these are clearly didactic and not intended to inform and delight the general reader. The present book still has an audience, but that audience is bright and curious high-school students and undergraduates rather than the general reader.

A good alternative or complement is Ore’s even older Number Theory and its History. That book has similar coverage and prerequisites, although it is more systematic, includes a lot of history, and has many more numerical examples. Another classic whose approach is very similar to the present book is Courant & Robbins & Stewart’s What is Mathematics?, which includes a moderate amount of number theory. Like the present book it displays a lot of equations and mathematical ideas, and like the present book is structured so that if you don’t understand something you can just skim along until you find something you do understand.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

Date Received: 
Saturday, April 11, 2015
Include In BLL Rating: 
C. Stanley Ogilvy and John T. Anderson
Publication Date: 
Allen Stenger
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The beginnings
2 Number patterns
3 Prime numbers as building blocks
4 Congruence arithmetic
5 Irrationals and iterations
6 Diophantine equations
7 Number curios
8 Prime numbers as leftover scrap
9 Calculating prodigies and prodigious calculations
10 Continued fractions
11 Fibonacci numbers
Publish Book: 
Modify Date: 
Thursday, November 11, 2010
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