Number Theory problems are among the most tricky in Mathematical Olympiads (MO). For students who are going to participate in such a tournament, and also for their teachers, a book that covers the main topics of fundamental number theory and contains various problems related to MOs is very useful. The book under review is exactly such a friendly volume, arranged in two main parts: Topics and Problems.
The author first reviews the fundamental topics, with some few numerical examples, but no exercises. So an amateur student needs to do some additional works to understand main results of topics. The author introduces the fundamental topics with complete proofs. Then follow 105 problems with detailed solutions. A number of them are selected from various MOs, and some of them are proposed mainly for MOs. Some of them have appeared in books before, but a large number of them are fresh.
The authors have chosen to cover many, but not all, topics of fundamental number theory. More precisely, in some chapters the author goes into in topics such as the work of Chebyshev on the approximation of the prime counting function π(x), and also studies the Riemann zeta function (mainly as a real function). The author doesn’t hide his analytic preferences. In particular he gives a step-by-step analysis of Newman’s short proof of the Prime Number Theorem, which of course requires tools from complex analysis. I have not seen such analysis before in books at this level, so this is a positive feature. On the other hand, I am not sure that this topic is very useful for MO students.
The book under review is not the only book which focuses on olympiad problems in number theory, but because of its structure (containing topics and problems), it is also useful for teaching. I highly recommend this book for students and teachers of MOs.
Mehdi Hassani is a faculty member at the Department of Mathematics, Zanjan University, Iran. His field of interest is Elementary, Analytic and Probabilistic Number Theory.
- Introduction.- The Fundamental Theorem of Arithmetic.- Arithmetic functions.- Perfect numbers, Fermat numbers.- Basic theory of congruences.- Quadratic residues and the Law of Quadratic Reciprocity.- The functions p(x) and li(x).- The Riemann zeta function.- Dirichlet series.- Partitions of integers.- Generating functions.- Solved exercises and problems.- The harmonic series of prime numbers.- Lagrange four-square theorem.- Bertrand postulate.- An inequality for the function p(n).- An elementary proof of the Prime Number Theorem.- Historical remarks on Fermat’s Last Theorem.- Bibliography and Cited References.- Author index.- Subject index.