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Number Theory Revealed: A Masterclass by Andrew Granville AMERICAN MATHEMATICAL SOCIETY, 2019, XXVIII+587 PP., $99.00, ISBN 978-1-4704-4158-6 REVIEWED BY MARCO ABATE

hat? Yet more introductory books on number theory? Aren’t there enough of them already? And why are we discussing them here? This is the Mathematical Intelligencer, not Mathematical Reviews! Indeed, the simple fact that this review is appearing here is a first hint that we have got something special. For starters, we are not presenting just one book; there are two, or three, or four, or even five of them, depending on how one counts (and counting is very important in number theory, as you know). Indeed, the pair of books under review here is intended to be followed by two (or three) other books—The Distribution of Primes: Analytic Number Theory Revealed and Rational Points on Curves: Arithmetic Geometry Revealed—that will present in detail deeper results in modern number theory (and they will be followed by Gauss’s Disquisitiones Arithmeticae Revealed, a modern reworking of Gauss’s classic book, one of the most influential texts in number theory). Number Theory Revealed is a series of books intended to constitute a first introduction to number theory, giving a survey of the subject starting from the very beginning and proceeding up to some glimpses of contemporary research. The Introduction is a condensed version of the Masterclass, containing just what is needed for a first course in number theory; but if you have even the slightest interest in number theory, I strongly suggest that you go for the Masterclass, which in its almost 600 pages contains a wonderful wealth of material. But the interest of this book series goes beyond the included material and depends also on a number of particularly effective stylistic and structural choices made by the author. Usually, mathematical textbooks proceed in a linear way. They start from the basic definitions and

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progress in an orderly fashion to more advanced material via theorems, examples, more definitions, and sometimes exercises. The material is supposed to be read in the order presented in the book, with no digressions; possibly at the end one might have a couple of chapters independent of the others. Graphically, a standard textbook can be represented by a line, with maybe just a few branches at the end. The topological structure of this book is instead much more complex: every chapter ramifies into many appendices, variously interconnected with each other and with future (or past) chapters. Moreover, some chapters (not necessarily the last ones) are very interesting but somewhat optional, and can be skipped on a first reading. The overall structure is very rich but complicated; this is one of the main reasons for the existence of two versions of the book. As mentioned above, the Introduction contains only the essential material for a first course in number theory (but every chapter still has an appendix); the full richness of the chosen structure appears in the Masterclass, with up to nine appendice

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