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Elementary Number Theory and Its Applications

Kenneth H. Rosen
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

The special feature of this book is the wealth of applications. Most number theory books today discuss RSA encryption as an application, but their hearts are not in it and it is something of a token. The present book takes it very seriously and has a whole chapter on modern encryption methods that depend on number theory. There are even discussions of attacks on RSA (codebreaking) that are based on number theory. It also has discussions of round-robin scheduling, splicing telephone cables to avoid crosstalk, hash searches, and pseudorandom number generation.

Curiously, in many ways Rosen’s book resembles Ore’s Number Theory and Its History (which Rosen’s book refers to frequently). It would be too much of a stretch to view Rosen’s book as a modern replacement for Ore’s, but they have the same leisurely pace, emphasis on numerical examples, and generous servings of history and biography. But Rosen’s book is packaged very much as a mainstream textbook, with carefully graduated exercises (and answers to all the odd-numbered ones included), a companion web site, and abundant instructor materials; while Ore’s was intended to be read for pleasure. In fact Rosen’s book may be a little boring to the teacher and to brighter students, because (except for the applications) it has the same proofs and developments as dozens of other number theory books, and it takes 750 pages to cover the beginning of number theory.

This Sixth Edition is not a drastic revision of the Fifth Edition; in fact the tables of contents are almost identical. The revisions are to bring the material up to date and to refine the book in general. There is a new section on partitions and a new section on congruent numbers (integers that are the area of a right triangle with rational sides) that is used as an introduction to elliptic curves.

One of the stated changes in this edition is “More attention than ever before has been paid to ensuring the accuracy of this edition.” (p. xi) This seems like a peculiar thing to claim as a feature, unless the previous editions were notably error-ridden, and it seems even more peculiar here because it appears in a section with the misspelled title “Accurancy”. I spotted several errors in the history and biography, but the mathematics seems to be error-free.

A Very Good Feature is the approximately 70 sidebars throughout the book giving short biographies of notable number theorists throughout history, from antiquity up through Terence Tao (born in 1975). Such a feature is valuable for reminding students that mathematics is made by humans, and in fact is still being made today. On the down side, the biographies are a little flimsy, there are no sources given, and they over-emphasize unverifiable anecdotes.

The obvious competitor for this book is Niven & Zuckerman & Montgomery’s An Introduction to the Theory of Numbers (hereafter NZM), even though that book is 20 years old. The books cover roughly the same topics, although NZM has a number of advanced topics not in Rosen and has a thorough treatment of elliptic curves. Rosen’s book is aimed lower; it assumes nothing beyond college algebra, while NZM uses some abstract algebra and real analysis. Rosen’s exercises are primarily numerical with a few proofs, while NZM’s exercises are primarily proofs with a few numerical examples, and in fact many additional interesting results are in NZM’s exercises. Rosen’s exposition gives a prominent place to the use of computers in number theory, and there are summaries of the leading commercial mathematics programs and packages in an appendix, while NZM barely mentions computers.

Bottom line: a modern, competent, and refined textbook that can be used for a variety of introductory courses, and having very thorough coverage of applications, but slow-paced and not very innovative.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

P. What is Number Theory?

1. The Integers.


Numbers and Sequences.


Sums and Products.


Mathematical Induction.


The Fibonacci Numbers.

2. Integer Representations and Operations.


Representations of Integers.


Computer Operations with Integers.


Complexity of Integer Operations.

3. Primes and Greatest Common Divisors.


Prime Numbers.


The Distribution of Primes.


Greatest Common Divisors.


The Euclidean Algorithm.


The Fundemental Theorem of Arithmetic.


Factorization Methods and Fermat Numbers.


Linear Diophantine Equations.

4. Congruences.


Introduction to Congruences.


Linear Congrences.


The Chinese Remainder Theorem.


Solving Polynomial Congruences.


Systems of Linear Congruences.


Factoring Using the Pollard Rho Method.

5. Applications of Congruences.


Divisibility Tests.


The perpetual Calendar.


Round Robin Tournaments.


Hashing Functions.


Check Digits.

6. Some Special Congruences.


Wilson's Theorem and Fermat's Little Theorem.




Euler's Theorem.

7. Multiplicative Functions.


The Euler Phi-Function.


The Sum and Number of Divisors.


Perfect Numbers and Mersenne Primes.


Mobius Inversion.

8. Cryptology.


Character Ciphers.


Block and Stream Ciphers.


Exponentiation Ciphers.


Knapsack Ciphers.


Cryptographic Protocols and Applications.

9. Primitive Roots.


The Order of an Integer and Primitive Roots.


Primitive Roots for Primes.


The Existence of Primitive Roots.


Index Arithmetic.


Primality Tests Using Orders of Integers and Primitive Roots.


Universal Exponents.

10. Applications of Primitive Roots and the Order of an Integer.


Pseudorandom Numbers.


The EIGamal Cryptosystem.


An Application to the Splicing of Telephone Cables.

11. Quadratic Residues.


Quadratic Residues and nonresidues.


The Law of Quadratic Reciprocity.


The Jacobi Symbol.


Euler Pseudoprimes.


Zero-Knowledge Proofs.

12. Decimal Fractions and Continued.


Decimal Fractions.


Finite Continued Fractions.


Infinite Continued Fractions.


Periodic Continued Fractions.


Factoring Using Continued Fractions.

13. Some Nonlinear Diophantine Equations.


Pythagorean Triples.


Fermat's Last Theorem.


Sums of Squares.


Pell's Equation.
Congruent Numbers.

14. The Gaussian Integers.


Gaussian Primes.


Unique Factorization of Gaussian Integers.


Gaussian Integers and Sums of Squares.