Number Theory is famous for problems that are easy to understand, but hard to solve. Lots of problems are still open and unsolved, and every year a large number of nice problems arise in mind of mathematicians. As its title shows, the aim of this book is to collect some of these unsolved problems.
There is no doubt that if one wants to keep a volume of this kind user-friendly it is not possible to cover all topics in number theory. Consequently, the author focuses on certain subjects, including elementary number theory, analytic and combinatorial number theory, and Diophantine equations. In short, the focus of book is on the main subjects of interest of P. Erdös, who was one of the great problem posers in Number Theory. One can find his name among the list of references at the end of many sections. On the other hand, very little is said of other areas, such as algebraic number theory or automorphic forms.
The book consist six chapters: A. Prime Numbers, B. Divisibility, C. Additive Number Theory, D. Diophantine Equations, E. Sequences of Integers, F. None of Above (quickly covering various problems, but not insisting on a special area). Each chapter consists of various sections, each of them studying a named problem. The author reviews previous work and gives the open problems. Then he gives a very rich list of references at the end of each section. Considering these lists, and also the fact that the author reviews current knowledge about each of the problems, this book is more than a problem book. Indeed, it is a book from which to start a serious study in Number Theory. It can be useful to professors, researchers and graduate students who want to find a subject for investigation, for undergraduate students and instructors who want to follow the subject and perhaps do some fully-meaningful computational experiments.
As the author mentions, in comparison with the first and second editions, the discussion of most of topics has been expanded, and a few new topics introduced as well. But because creating new problems and daily getting closer to finding solutions to some of them is what mathematicians do, this book will always need new editions.
Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics, Analytic Number Theory in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.