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Number Theory: An Introduction via the Density of Primes

Benjamin Fine and Gerhard Rosenberger
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is an interesting take on an introductory number theory course, one that is heavily slanted toward the prime numbers. As such, it omits or mentions only briefly many topics that would normally appear in an introductory course, such as Diophantine equations, partitions, and continued fractions. It squeezes in a little bit on irrational numbers and on quadratic reciprocity, but these are given as isolated results that don’t go anywhere.

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. The prerequisites are higher than usual for a book at this level: single and multiple variable calculus and linear algebra. The book develops all the complex analysis and abstract algebra it needs.

The book is subtitled “An Introduction via the Density of Primes”, which is not really true because it does many other things too. It does have very good coverage of prime distribution, including proofs of Dirichlet’s theorem on primes in arithmetic progressions, Bertrand’s Postulate that between each number and its double is a prime, Newman’s simple analytic proof of the prime number theorem, a mostly-complete sketch of the Selberg–Erdös elementary proof of the prime number theorem, and Brun’s theorem that the sum of the reciprocals of the twin primes converges. There’s a lot on primality testing, including a description of the AKS polynomial-time algorithm. Most of the proofs are familiar ones; the innovation in the book is in the choice of topics rather than the approach.

The emphasis on primes leads to some peculiarities, especially because of the exclusion of Diophantine equations. For example, there are lengthy chapters on algebraic numbers and on p-adic numbers that has almost no applications. There’s also a good development of elliptic curves, but they are used only for primality testing and proving and not for studying equations.

Bottom line: a good text for an introductory course, if you have enough flexibility to omit many traditional topics.

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

See the table of contents in the publisher's webpage.