This book discusses various problems in elementary number theory, most of which have a combinatorial flavor. In general classical problems are avoided and almost no proofs are given for the presented problems. Both the difficulty and importance of the problems discussed are very variable in some are only exercises while others are very difficult or even hopeless and may have important consequences or their eventual solution may lead to important advances and the discovery of new methods. This new edition is the joint work of the late Paul Erdys, Ron Graham, and as new co-authors, Melvin Nathanson and Xingde Jia.