## The Classical Bases

• Melvyn B. Nathanson
Book

Part of the Graduate Texts in Mathematics book series (GTM, volume 164)

1. Front Matter
Pages i-xiv
2. ### Waring’s problem

1. Front Matter
Pages 1-1
2. Melvyn B. Nathanson
Pages 3-36
3. Melvyn B. Nathanson
Pages 37-74
4. Melvyn B. Nathanson
Pages 75-95
5. Melvyn B. Nathanson
Pages 97-119
6. Melvyn B. Nathanson
Pages 121-148
3. ### The Goldbach conjecture

1. Front Matter
Pages 149-149
2. Melvyn B. Nathanson
Pages 151-175
3. Melvyn B. Nathanson
Pages 177-209
4. Melvyn B. Nathanson
Pages 211-230
5. Melvyn B. Nathanson
Pages 231-270
6. Melvyn B. Nathanson
Pages 271-298
4. Back Matter
Pages 299-344

### Introduction

[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.

### Keywords

Waring's problem integral number theory real analysis

#### Authors and affiliations

• Melvyn B. Nathanson
• 1
1. 1.Department of MathematicsLehman College of the City University of New YorkBronxUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-3845-2
• Copyright Information Springer-Verlag New York 1996
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4419-2848-1
• Online ISBN 978-1-4757-3845-2
• Series Print ISSN 0072-5285
• Buy this book on publisher's site