An Open Door to Number Theory

This is an idiosyncratic introductory text for a one-semester number theory course. It’s not comprehensive, even compared to other brief introductions. Instead it weaves a path through various interesting topics and shows how they are connected and can be extended. The prerequisites are minimal; a little bit of abstract algebra would be helpful, but a good grasp on how to prove things is more important. There is a complete solutions manual available in electronic format, but I did not examine this.

An example of the weaving and wandering is that continued fractions occupy about 25 pages in the first chapter. This topic is rarely introduced so early, but it works here. The exposition boldly introduces the recurrences for calculating the convergents (under the name “the amazing array”) and simply starts applying it without justifying it. The exposition ties together the Euclidean algorithm, solutions of linear equations, approximation of irrationals by rationals, algorithms, and recurrences, all without being overwhelming or too abstract. Then finally the book doubles back and proves the correctness of “the amazing array”.

One interesting and valuable feature of the book is that it introduces the ideas of rings and fields right away, and spends a good deal of time not only in the ring of integers but in the “smaller” rings (integers modulo \(m\)) and “larger” rings (field extensions by adjoining roots of equations). I think this improves understanding by emphasizing which properties of the integers carry over to other integers-like structures and which do not. The treatment is kept simple and for the “larger” rings emphasizes particular examples rather than proving general theorems about field extensions. There’s also an optional project at the end that gives a brief development of \(p\)-adic numbers. The book is slanted towards algebraic number theory, although in the particular problems of representing numbers by quadratic forms rather than for Diophantine equations or general number fields.

Very Good Feature: The last fifth of the book is devoted to several semester projects, each covering a more advanced topic. Each has a brief exposition part and then a series of fairly difficult problems for the student to work on. The author assigns three or four of these in a semester course.

One unusual feature is that the book looks at many tables of data and invites the reader to find patterns and guess theorems. (It’s not an inquiry-based book, as it does state and prove a lot of theorems, but it puts a lot of the development into the exercises and invites the reader to make his own discoveries.) Another book that takes a data-driven approach is Burn’s similarly-titled A Pathway into Number Theory. Burn takes a purer approach, where nearly everything is driven by examining patterns in tables of data, while in Campbell this is an important but not central technique. Neither book mentions using computers to generate the data; this is more surprising for Campbell, which was written in 2018, than in Burn, written in 1982.

Another idiosyncratic book, that also takes a wandering and weaving approach, is Shanks’s Solved and Unsolved Problems in Number Theory. This is a much more advanced book and much denser, although it too starts at the beginning and uses lots of examples and tables of data to drive the narrative. I believe Shanks’s book is not a good introduction, but it’s an interesting development of number theory.

Bottom line: An Open Door to Number Theory is a very interesting and leisurely introduction. Because of its uneven coverage, I think it works best as a capstone course or an introduction to algebraic number theory. It is a good choice if you like the topics covered.

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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 allenstenger.com. His mathematical interests are number theory and classical analysis.