Preface xi which provide a better understanding of the multiplicative structure of the integers. For instance, we study the average value of the number of prime factors of an integer, the average value of the number of its divisors, the behavior of its smallest prime factor and of its largest prime factor, and so on. A whole chapter is devoted to sieve methods, and many of their applications are presented in the problem section at the end of that chapter. Moreover, we chose to include some results which are hard to find elsewhere. For instance, we state and prove the very useful Birkhoff-Vandiver primitive divisor theorem and the important Tur´ an-Kubilius inequality for additive functions. We also discuss less serious but nevertheless interesting topics such as the Erd˝ os multiplication table problem. We also chose to discuss the famous abc conjecture, because it is fairly recent (it was first stated in 1985) and also because it is central in the study of various conjectures in number theory. Finally, we devote a chapter to the study of the index of composition of an integer, its study allowing us to better understand the anatomy of an integer. To help the reader better comprehend the various themes presented in this book, we listed 263 problems along with the solutions to the even- numbered ones. Finally, we are grateful to our former students who provided important feedback on earlier versions of this book. In particular, we would like to thank Maurice-Etienne Cloutier, Antoine Corriveau la Grenade, Michael Daub, Nicolas Doyon, Ross Kravitz, Natee Pitiwan, and Brian Simanek. We are very appreciative of the assistance of Professor Kevin A. Broughan, who kindly provided mathematical and grammatical suggestions. We would also like to thank the anonymous reviewers of the AMS for their clever suggestions which helped improve the quality of this book. Jean-Marie De Koninck Florian Luca
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