2009;
303 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-4880-7

Product Code: MBK/68

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-1207-4
Product Code: MBK/68.E**

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

#### Supplemental Materials

# Not Always Buried Deep: A Second Course in Elementary Number Theory

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*Paul Pollack*

Number theory is one of the few areas of mathematics where problems of
substantial interest can be fully described to someone with minimal
mathematical background. Solving such problems sometimes requires
difficult and deep methods. But this is not a universal phenomenon; many
engaging problems can be successfully attacked with little more than one's
mathematical bare hands. In this case one says that the problem can be
solved in an elementary way. Such elementary methods and the problems to
which they apply are the subject of this book.

Not Always Buried Deep is designed to be read and enjoyed
by those who wish to explore elementary methods in modern number
theory. The heart of the book is a thorough introduction to
elementary prime number theory, including Dirichlet's theorem on
primes in arithmetic progressions, the Brun sieve, and the
Erdös–Selberg proof of the prime number theorem. Rather
than trying to present a comprehensive treatise, Pollack focuses on
topics that are particularly attractive and accessible. Other topics
covered include Gauss's theory of cyclotomy and its applications to
rational reciprocity laws, Hilbert's solution to Waring's problem, and
modern work on perfect numbers.

The nature of the material means that little is required in terms of
prerequisites: The reader is expected to have prior familiarity with
number theory at the level of an undergraduate course and a first course
in modern algebra (covering groups, rings, and fields). The exposition is
complemented by over 200 exercises and 400 references.

#### Readership

Undergraduates, graduate students, and research mathematicians interested in number theory.

#### Reviews & Endorsements

This book provides a gentle introduction to more advanced number theory. The mathematics is intricate at times, but not unnecessarily so. One gets the sense that Pollack took great care to make the material as accessible as possible. ... Pollack's book is pleasant to read. The book appears to have been typeset in LaTeX just like countless other math books, but something about it makes the print more attractive. Perhaps it is the paper color; the pages have a subtle parchment tone. The book is also pleasant to read for its elegant mathematical content.

-- MAA Reviews

...one of the best mathematics books that I have read recently. It is beautifully written and very well organized, the kind of book that is well within the reach of an undergraduate student, even one with little complex analysis. Indeed, a good knowledge of the analysis of real functions of one variable is probably enough for reading most of the book. ... I know of no better place to learn about Dirichlet's Theorem on arithmetic progressions or Selberg's proof of the Prime Number Theorem. And if these are two results of analytic number theory that deserve to be known to every mathematician, these are certainly they.

-- S. C. Coutinho

Paul Pollack's book...presents a specially beautiful selection of topics in Elementary Number Theory. By this we mean that all of the problems addressed can be—and, in fact, are—described in simple terms and that the mathematics involved are, for the most part, self-sufficient. ... [The] refreshing point of view makes this book specially suitable for professional mathematicians not specialists on the subject who want to learn about contemporary Elementary Number Theory, for number theorists who want to keep up with the state of the art in the subject or just give themselves the pleasure of reading a beautiful book and as supplementary material in a second level course on Elementary Number Theory.

-- Capi Corrales Rodrigaqez, EMS Newsletter

This very interesting, well-written book is both enjoyable and informative for those with the appropriate background. ... The mathematics is done clearly with quotes from the "masters" interspersed at appropriate points. ... Highly recommended.

-- CHOICE Magazine

This is an excellent introductory book to analytic number theory. It is ideal for a first course in analytic number theory at the undergraduate level.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Not Always Buried Deep: A Second Course in Elementary Number Theory

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Foreword xi12 free
- Elementary prime number theory, I 118 free
- Cyclotomy 4562
- Elementary prime number theory, II 85102
- Primes in arithmetic progressions 119136
- Interlude: a proof of the Hilbert–Waring theorem 151168
- Sieve methods 163180
- An elementary proof of the prime number theorem 213230
- Perfect numbers and their friends 247264
- References 279296
- Index 301318 free
- Back Cover Back Cover1322