The book of numbers. Translated from the Chinese by the author and Jiu Ding (Q2818808)

This is a gem of a book. When I received this book for review, I showed it to my neighbour, a talented high school student interested more in English literature than in mathematics. However, she was fascinated by the narrative style and enjoyed reading the book. That tells me that the book can reach a wide audience. Today, when I am typing this review, it is the world poetry day and the author is also a poet renowned in his country.NEWLINENEWLINEThere have always been popular books written on elementary number theory as it is a subject where many fascinating questions can be asked that can be understood by the proverbial layman but whose solutions are deep in reality. The author being a number theorist, has crafted meticulously a book free of errors while communicating the excitement that he obviously feels for the subject.NEWLINENEWLINEThe first chapter which discusses perfect numbers, Mersenne primes and aspects of the division algorithm has already some novel contents not found in a typical book on elementary number theory. Interspersed there is a description of tiling geometry and the Euler characteristic. The author also discusses Graham's conjecture and Sylvester's theorem (a generalization of Bertrand's postulate). While describing the twin prime conjecture, he mentions generalizations as well as the latest known results of \textit{Y. Zhang} [Ann. Math. (2) 179, No. 3, 1121--1174 (2014; Zbl 1290.11128)] and \textit{B. Green} and \textit{T. Tao}'s work [Ann. Math. (2) 175, No. 2, 465--540 (2012; Zbl 1251.37012); erratum ibid. 179, No. 3, 1175--1183 (2014); ibid. (2) 175, No. 2, 541--566 (2012; Zbl 1347.37019)]. Only an expert number-theorist would know the extent to which state-of-the-art results could be described.NEWLINENEWLINEThe second chapter is on congruences, a notion (and notation too) introduced by the great mathematician Gauss. There are some new features here -- apart from a nice discussion of RSA cryptosystem, the author also analyses the Jordan totient function. He also discusses the decimal expansion of rational numbers pointing out that the length of the period is the order of 10 modulo the denominator; he misses out on the opportunity to define the notion of primitive root modulo primes.NEWLINENEWLINEIn Chapter 3, the author is understandably proud to introduce the Chinese remainder theorem in its original name. An interesting topic discussed here is that of covering congruences. The 4th chapter contains a study of the important quadratic reciprocity law of Gauss. Here again two novel inclusions are Hadamard matrices and a construction of the 17-gon. The 5th chapter looks at higher power residues. It also talks about Egyptian fractions and also discusses Artin's conjecture on primitive roots modulo primes in some detail. It would have been ideal to mention the ``\textit{Chakravala}'' method while discussing the so-erroneously-called Pell's equation. Indeed, the import of the work of Brahmagupta and Bhaskara on Pell's equation can be divined from the 20th century great André Weil's comment: ``What would have been Fermat's astonishment if some missionary, just back from India, had told him that his problem had been successfully tackled there by native mathematicians almost six centuries earlier ?'' This was in reference to Fermat, who in 1657, writing to his friend Frenicle, posed ``to the English mathematicians and all others'' the problem of finding a solution of \(x^2-Ny^2=1\) ``pour ne vous donner pas trop de peine'' like \(N=61, 109\).NEWLINENEWLINEChapter 6 is rich with a lot of beautiful number theory. Bernoulli numbers, their relation to the Riemann zeta function, partition functions and elliptic curves are briefly discussed. There is also a mention of the \(abc\) and Catalan conjectures. Finally, Chapter 7 studies among other things Waring's problem and the congruent number problem. The author also describes some of his work on Diophantine equations and on a generalization of the congruent number problem. These last two chapters have a lot of rich mathematics.NEWLINENEWLINEIf one were to point out shortcomings, I would say: (i) an index/glossary of terms is missing, (ii) P. Ribenboim's excellent books on prime number records should have been mentioned, and (iii) the ``\textit{Chakravala}'' method is too important not to be mentioned while discussing the Brahmagupta-Pell equation. However, the several noteworthy features of the book noted above more than compensate for these. I expect professional mathematicians as well as students interested in mathematics will enjoy reading the book. I certainly did.