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