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[Hilbert's] style has not the terseness of many of our modem authors in mathematics, which is based on the assumption that printer's labor and paper are costly but the reader's effort and time are not. H. Weyl [143] The purpose of this book is to describe the classical problems in additive number theory and to introduce the circle method and the sieve method, which are the basic analytical and combinatorial tools used to attack these problems. This book is intended for students who want to lel?Ill additive number theory, not for experts who already know it. For this reason, proofs include many "unnecessary" and "obvious" steps; this is by design. The archetypical theorem in additive number theory is due to Lagrange: Every nonnegative integer is the sum of four squares. In general, the set A of nonnegative integers is called an additive basis of order h if every nonnegative integer can be written as the sum of h not necessarily distinct elements of A. Lagrange 's theorem is the statement that the squares are a basis of order four. The set A is called a basis offinite order if A is a basis of order h for some positive integer h. Additive number theory is in large part the study of bases of finite order. The classical bases are the squares, cubes, and higher powers; the polygonal numbers; and the prime numbers. The classical questions associated with these bases are Waring's problem and the Goldbach conjecture.
 Reviews

From the reviews:
“This book provides a very thorough exposition of work to date on two classical problems in additive number theory … . is aimed at students who have some background in number theory and a strong background in real analysis. A novel feature of the book, and one that makes it very easy to read, is that all the calculations are written out in full – there are no steps ‘left to the reader’. … The book also includes a large number of exercises … .” (Allen Stenger, The Mathematical Association of America, August, 2010)
 Table of contents (10 chapters)


Sums of polygons
Pages 336

Waring’s problem for cubes
Pages 3774

The HilbertWaring theorem
Pages 7595

Weyl’s inequality
Pages 97119

The Hardy—Littlewood asymptotic formula
Pages 121148

Table of contents (10 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Additive Number Theory The Classical Bases
 Authors

 Melvyn B. Nathanson
 Series Title
 Graduate Texts in Mathematics
 Series Volume
 164
 Copyright
 1996
 Publisher
 SpringerVerlag New York
 Copyright Holder
 SpringerVerlag New York
 eBook ISBN
 9781475738452
 DOI
 10.1007/9781475738452
 Hardcover ISBN
 9780387946566
 Softcover ISBN
 9781441928481
 Series ISSN
 00725285
 Edition Number
 1
 Number of Pages
 XIV, 342
 Topics