You are here

The Book of Numbers

Tianxin Cai
World Scientific
Publication Date: 
[Reviewed by
Mark Hunacek
, on

This book, a translation from the Chinese, is intended as a text for a course in elementary number theory at the upper undergraduate/lower graduate level. While it has some interesting features, there are also aspects of this book that, I think, would make its use as a text problematic.

First the good news. The table of contents (for which, click the link above) does not fully indicate the breadth of topics covered in the text. The reader of this book will encounter, in addition to the basics of divisibility and congruences (including quadratic reciprocity), some discussion of, among other things: Gaussian integers, Waring’s problem, the abc conjecture, Graham’s conjecture, constructibility of the 7-gon and 17-gon, continued fractions, Dirichlet characters and modular forms. In addition, very recent developments are included for many topics, including original work.

The book also places more of an emphasis than usual on Chinese contributions, both old and new, to number theory, thereby serving the salutary function of reminding students that not all mathematics has been created by Americans and Europeans. As an example, the standard result typically referred to as the Chinese Remainder Theorem is here called the Qin Jiushao theorem to honor the man who, according to the author, “made the greatest contribution to this theorem.” The author also quotes a Chinese mathematician, Pan Chengdong, as believing that “the reason why westerners named the above theorem in their way is that, in addition to the difficult pronunciation of a Chinese name, this is the only great theorem obtained by Chinese mathematicians. So it is a kind of prejudice.” Whether one agrees or disagrees with this conclusion, it is an interesting glimpse into a point of view held by some people of another culture.

On the other hand, this emphasis on Chinese contributions can at times seem overdone, especially since the author makes repeated reference to his own work. I counted at least 25 times (I mean this literally) where the author mentioned his own contributions, each mentioned accompanied by some use of the first person, as in “we proved”, “we generalized”, “we proposed”, etc. On one occasion he refers to his 1984 master’s thesis, and on another he mentions that he once conjectured a result that had been conjectured years earlier by Murty: “In 2012, the author, who did not notice the work of Murty, proposed Conjecture 7.5….”. These repeated references give the book a definite, even if unintentional, air of self-promotion.

This is one negative feature of the book. Another is the exercise sets, which struck me as very skimpy. Exercises appear at the end of the first five (but not the last two) chapters; the number of exercises per chapter ranges from 11 to 13. That’s 59 exercises for an entire text; by contrast, An Introduction to the Theory of Numbers by Niven, Zuckerman and Montgomery contains almost that many (54) in its first substantive section, on divisibility.

The fact that this book is a translation from another language is also often apparent. The writing lacks the elegance and readability of, say, Silverman’s A Friendly Introduction to Number Theory, and the English is occasionally quite idiosyncratic, as in “We made a research…” or “The gold chicken is getting fewer and fewer.” Additionally, the book’s references to Sophie Germaine as “Sophie”, a practice not employed for male mathematicians, might well strike some people as offensive and condescending.

The order of presentation of some of the topics also did not seem optimal to me for an introductory text. In a book like this, one would typically expect to see, as the first topics covered, the very basics of divisibility theory and prime factorization. In this book, however, before the author has even discussed the fundamental theorem of arithmetic, he has spent several pages discussing Graham’s conjecture, a topic that, although interesting, seems somewhat tangential to an introduction to the subject, especially this early.

This book is clearly a useful desk reference for instructors teaching a course in number theory. However, given the fact that there are already quite a few number theory textbooks available, some of them (like the two mentioned earlier) very good, I can’t say that this book represents a pedagogical advance on the current literature.

Mark Hunacek ( teaches mathematics at Iowa State University. 

  • The Mystery of Natural Numbers
  • The Concept of Congruence
  • Congruence
  • Quadratic Residue
  • The nth Power Residues
  • Congruence Modulo Integer Power
  • Additive and Multiplicative Number Theory