An Introduction to the Theory of Numbers

Reviewed by Clayton Lungstrum, Instructor, Virginia Tech on 12/22/21

Comprehensiveness rating: 4 see less

Moser's book covers a wide variety of topics, and in that sense, it is comprehensive. However, each section is rather short, so while it touches on a lot of topics, it does not dive very deeply into any single topic.

Content Accuracy rating: 5

Moser's book has no mistakes in it that I found.

Relevance/Longevity rating: 4

Mathematics is somewhat unique among the sciences: the text itself was written in the 1950s but it is still very applicable in undergraduate classrooms today. For a class that isn't focused on any one particular area of number theory but rather aiming to provide a broad overview of multiple areas of research, this books does an excellent job at bringing together the topics and presenting them in a cohesive manner that is likely to be relevant for a long time to come.

Clarity rating: 4

This is certainly a traditional math text that requires a pen and paper when you sit down to read through it. Moser often gives a clear indication of how to work through the proofs of more difficult theorems, but he doesn't take the time to work out all of the calculations. It is in this sense that his book is written very well, though there are some instances where I found his exposition to be a little confusing and it took me some time to understand his approach.

Consistency rating: 5

The terminology in this textbook has largely become commonplace in modern mathematical literature and it is used consistently throughout the book without confusing any meanings.

Modularity rating: 5

Each section typically spans two-to-three pages, depending on the calculations involved for each proof. This makes it very easy to determine which sections are suitable for students to read on their own versus being presented in class.

Organization/Structure/Flow rating: 3

The organization of the book is neither good nor bad; some of the chapters follow the ``obvious'' chapters (such as arithmetic functions preceding the distribution of primes). However, some of the topics feel unrelated to preceding chapters (such as irrational numbers following the distribution of primes chapter). Each chapter in and of itself is well organized and flows well, but I think that the chapters themselves have potential to be reorganized to present a logical order for the textbook as a whole.

Interface rating: 5

All of the formulas display clearly and as expected throughout the text.

Grammatical Errors rating: 5

The writing is clean and clear throughout the textbook. The sentences do not needlessly drone on, and the details included are frequently very carefully phrased so that each word has the greatest impact possible without sacrificing space.

Cultural Relevance rating: 5

This book is not offensive or insensitive in any way; this is a clear explanation of a wide variety of number theoretic topics.

Comments

The exercises presented in this book are superb. Moser has gone above and beyond and also included a list of unsolved problems/conjectures that can assist readers that enjoy their time spent reading this book finding places to begin research. As a teacher of undergraduate classes, I find this to be a refreshing take on traditional math texts, removing the ``plug-and-chug'' nature of many math courses and replacing it with an exploratory element that is the very essence of much of mathematics.