The second edition of Manin and Panchiskin's introductory volume on Number Theory for Springer's Encyclopaedia of the Mathematical Sciences series is about twice the size of the first. That in itself is already good news; the first edition was a very good book; this edition is even better.
The book is intended as a sort of survey of all of modern number theory, while at the same time serving as the introduction to the other number theory volumes in this series. The style is typical of broad surveys: big theorems are quoted carefully but not proved, but simple arguments and proofs are often given when they don't require a lot of space. Embedded in the text are a lot of interesting ideas, insights, and clues to how the authors think about the subject.
The first part, entitled "Problems and Tricks", has two chapters. The first runs through most of the standard topics in elementary number theory, from primes and divisibility to diophantine equations and diophantine approximations. Following that survey, one gets a chapter on "applications of number theory", which is really almost entirely about computational number theory (primality tests, factorization), which count as "applications" because of their connection to public-key cryptography. New to this edition is a description of the AKK algorithm for primality testing in polynomial time.
Things get more interesting in Part II (by far the largest of the three parts), entitled "Ideas and Theories." This is provocative, of course, since many would be surprised to be told there were no "ideas" in elementary number theory, only "tricks". The authors' point is that while the elementary methods often seem ad hoc, number theory is now understood mostly in the light of broad ideas and highly sophisticated theories, from which the most important recent advances have sprung. This part of the book covers such things as approaches through logic, algebraic number theory, arithmetic of algebraic varieties, zeta functions, and modular forms, followed by an extensive (50+ pages) account of Wiles' proof of Fermat's Last Theorem. This is a valuable addition, new in this edition, and serves as a vivid example of the power of the "ideas and theories" that dominate this part of the book.
Also new and very interesting is Part III, entitled "Analogies and Visions," which proposes that the way forward for number theory lies in Arakelov Geometry and Noncommutative Geometry. This is very much a personal point of view of the authors, one whose merits are still unclear. Of course, that is what "visions" are all about!
The best surveys of mathematics are those written by deeply insightful mathematicians who are not afraid to infuse their ideas and insights into their outline of the subject. This is what we have here, and the result is an essential book. I only wish the price were lower so that I could encourage my students buy themselves a copy. Maybe I'll do that anyway.
Fernando Gouvêa is the author of p-adic Numbers: An Introduction.