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Not Always Buried Deep: A Second Course in Elementary Number Theory
Paul Pollack, University of Illinois, Urbana-Champaign, IL
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2009; 303 pp; hardcover
ISBN-10: 0-8218-4880-1
ISBN-13: 978-0-8218-4880-7
List Price: US$66 Member Price: US$52.80
Order Code: MBK/68

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.

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Reviews

"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