From the author In my book I present a systematic exposition of the funda­ mentals of number theory within the scope of a university course. A large collection of problems introduces the reader to some of the new ideas in number theory.This fifth edition of my book differs considerably from the fourth. A series of changes, allowing a simpler exposition, have been made in all the chapters of the book. The most important changes are the merging of the old chapters IV and V into one chapter IV (reducing the number of chapters to six) and the new, simpler proof of the existence of primitive roots.The problems at the end of each chapter have been essentially revised. The order of the problems is now in complete cor­ respondence with the order of the presentation of the theoretical material. Some new problems have been added; but the number of numbered problems has been substantially reduced. This was accomplished by the unification, under the letters a, b, c, . . ., of previously separate problems which were re­ lated by the method of solution or by content. All the solutions of the problems have been reviewed; in many cases these solu­ tions have been simplified or replaced by better ones. Particu­ larly essential changes have been made in the solutions of the problems relating to the distribution of n-th power residues and non-residues, and primitive roots, as well as in the estimations of the corresponding trigonometric sums.