The Preface to the First Edition A962) states that this is "a rather
tightly organized presentation of elementary number theory" and that
"number theory is very much a live subject." These two facts are in
conflict fifteen years later. Considerable updating is desirable at many
places in the 1962 text, but the needed insertions would call for drastic
surgery. This could easily damage the flow of ideas and the author was
reluctant to do that. Instead, the original text has been left as is, except
for typographical corrections, and a brief new chapter entitled "Pro-
"Progress" has been added. A new reader will read the book at two
levels—as it was in 1962, and as things are today.
Of course, not all advances in number theory are discussed, only those
pertinent to the earlier text. Even then, the reader will be impressed
with the changes that have occurred and will come to believe—if he did
not already know it—that number theory is very much a live subject.
The new chapter is rather different in style, since few topics are
developed at much length. Frequently, it is extremely brief and merely
gives references. The intent is not only to discuss the most important
changes in sufficient detail but also to be a useful guide to many other
topics. A propos this intended utility, one special feature: Developments
in the algorithmic and computational aspects of the subject have been
especially active. It happens that the author was an editor of Mathe-
Mathematics of Computation throughout this period, and so he was particu-
particularly close to most of these developments. Many good students and
professionals hardly know this material at all. The author feels an
obligation to make it better known, and therefore there is frequent
emphasis on these aspects of the subject.