Andrew Granville has written many wonderful expository articles about number theory (especially patterns in primes), as well as dozens of research articles on many aspects of this field. It is thus highly appropriate that he turn all this knowledge into a series of textbooks for number theory. The volume under review is the first.

Well, sort of. The grand plan of "Number Theory Revealed" is that it will be followed by two more volumes (one on the Prime Number Theorem and other analytic topics, the other on counting points on Mordell curves and geometry), plus a bonus volume retelling Gauss' *Disquisitiones* from Granville's perspective. If this sounds more ambitious than simply a textbook series, you would be right, and that characterizes this book, subtitled "The Masterclass", as well.

The first thing one must say about this book is that for anyone who is seriously interested in teaching a number theory course, much less has done so many times, it is ridiculously enjoyable to read. Every page has intriguing exercises, references to cutting-edge literature, or unsuspected topics even jaded instructors will smile at.

For instance, remember when the internet (well, the math internet) broke over writing 33 as a sum of just three cubes? It's here, with context. Did you know the famous Euler prime-producing polynomial \(x^2+x+41\) is related to the fact that -163 is the biggest negative discriminant of class number 1? I should have, but learned it here first.

That is not to say that the topics covered are nonstandard. In fact, the most general topics of the first 8 chapters (what Granville recommends for a minimalist course, for instance for future teachers) are extremely standard and, if terse, are accurate and within a long tradition of such number theory texts. From the fundamental theorem of arithmetic to the Chinese remainder theorem to primitive roots and quadratic reciprocity, it is here, in the usual order.

However, taking a deeper dive reveals that this book, at least in my view, would be pretty difficult to use for a moderate undergraduate course as-is without substantial supplementing. For instance, just 16 pages are given to quadratic residues, including 3 proofs of Euler's criterion, five proofs of when -1 is a residue, a two-page proof of quadratic reciprocity, and the use of the Jacobi symbol. While there are some examples (thankfully, three for using quadratic reciprocity), they are not set off typographically, and in far more cases are left as some of the copious in-text exercises or just left to the reader's diligence to think up.

Some minor choices are also surprising, at least to me. There seemed to be more about computational complexity than about the practice of cryptography. The historical notes are sometimes very careful, but omit Qin Jiushao (and everyone else) completely when describing the generalized Chinese remainder theorem. Can you really cover everything about partitions in two pages? (Amazingly, yes, but was it advisable?)

Those quibbles may reveal my own topical biases, but more substantive of a gripe is the editing, which seems rushed. The index is really short, for a nearly 600-page book, and the exercise hints are tantalizingly brief. (Exercises are available separately for free download.) I can find no reason there shouldn't have been a proper bibliography at the end instead of having most references scattered section by section. Inasmuch as the author pays homage to Hardy and Wright and expressly does not want to produce a comprehensive set of references that is fine, but I am sure that BibTeX could have at least just repeated all the actually used references again at the end pretty easily for ease of use.

On the other hand, there would be no reason to complain about references without the huge positive of this book; there are just so many amazing tidbits fleshed out in one place! Much discussion comes from his own previous expository articles (e.g. Bhargava composition of quadratic forms, Pascal's triangle modulo various numbers) but there are so many others. For instance, consider that Bertrand's Postulate/Chebyshev's Theorem on the existence of a prime between a number and its double is a standard topic in undergraduate texts (though usually not covered in just one page). Here, Appendix 5A goes on to introduce and prove the Sylvester-Schur theorem on prime divisors of more general integer ranges; this is exactly the kind of thing one could have students explore, whether they ever saw a proof - or even statement - of the original theorem.

As long as we keep an appropriate audience in mind, then things like the appendices to the residues chapter being longer than the chapter itself are just fine. (Another edition, *The Introduction*, cuts about half the material, making for a lean and standard, though terse, course.) Similarly, some exercises are straightforward, but as Granville mentions at one point, some others "get considerably more involved"; the double dagger (very hard!) exercise on binary quadratic forms referring directly to an appendix on Dirichlet L-functions comes to mind.

So if you are looking for enrichment in how the world of number theory connects together but formatted as a textbook with familiar topical arrangement, this is a great resource; I can't wait for the next two volumes to appear.

Karl-Dieter Crisman is Professor of Mathematics at Gordon College. When asked to come up with a course for majors as a new faculty member, he immediately jumped at the chance to teach number theory, which led to an abiding interest in using open source software to explore this deep subject with students in class.