Number Theory
An Introduction via the Density of Primes
Authors: Fine, Benjamin, Rosenberger, Gerhard
Free Preview A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem
 Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals
 One of the few books to include the AKS algorithm that shows that primality testing is one of polynomial time
 Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers
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 About this Textbook

Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of padic numbers, Hensel's lemma, multiple zetavalues, and elliptic curve methods in primality testing.
Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem
 Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals
 Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts
 Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers
 A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem
 About the authors

Benjamin Fine, PhD, is Professor of Mathematics at Fairfield University, CT, USA.
Gerhard Rosenberger, PhD, is Professor (retired) at Dortmund University of Technology, Germany.
 Reviews

“In this text, Fine (mathematics, Fairfield Univ.) and Rosenberger (Univ. of Hamburg, Germany) successfully present number theory from the inception of primes to recent developments in algebraic and analytic number theory and cryptography. … Numerous exercises and open problems are provided. The breadth and depth of topics covered are impressive, making this an excellent text for those interested in the field of number theory. Summing Up: Recommended. Upperdivision undergraduates and graduate students.” (J. T. Zerger, Choice, Vol. 54 (9), May, 2017)
“The book is chatty and leisurely, with lots of historical notes and lots of worked examples. The exercises at the end of each chapter are good and there are a reasonable number of them. … a good text for an introductory course … .” (Allen Stenger, MAA Reviews, maa.org, November, 2016)
 Table of contents (7 chapters)


Introduction and Historical Remarks
Pages 15

Basic Number Theory
Pages 758

The Infinitude of Primes
Pages 59142

The Density of Primes
Pages 143218

Primality Testing—An Overview
Pages 219284

Table of contents (7 chapters)
 Download Preface 1 PDF (63.4 KB)
 Download Sample pages 2 PDF (428.1 KB)
 Download Table of contents PDF (177.7 KB)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Number Theory
 Book Subtitle
 An Introduction via the Density of Primes
 Authors

 Benjamin Fine
 Gerhard Rosenberger
 Copyright
 2016
 Publisher
 Birkhäuser Basel
 Copyright Holder
 Springer International Publishing AG
 eBook ISBN
 9783319438757
 DOI
 10.1007/9783319438757
 Hardcover ISBN
 9783319438733
 Softcover ISBN
 9783319829319
 Edition Number
 2
 Number of Pages
 XIII, 413
 Number of Illustrations
 11 b/w illustrations, 1 illustrations in colour
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