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About this book
[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.
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Table of contents (10 chapters)
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Front Matter
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Waring’s problem
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Front Matter
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The Goldbach conjecture
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Front Matter
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Back Matter
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)Authors and Affiliations
Bibliographic Information
Book Title: Additive Number Theory The Classical Bases
Authors: Melvyn B. Nathanson
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-1-4757-3845-2
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer-Verlag New York 1996
Hardcover ISBN: 978-0-387-94656-6Published: 25 June 1996
Softcover ISBN: 978-1-4419-2848-1Published: 19 November 2010
eBook ISBN: 978-1-4757-3845-2Published: 14 March 2013
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XIV, 342
Topics: Number Theory, Analysis