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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