You are here

Additive Number Theory: The Classical Bases

Melvyn B. Nathanson
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics 164
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This book provides a very thorough exposition of work to date on two classical problems in additive number theory: Waring’s Problem, which asserts that, for each fixed positive integer n, every positive integer is the sum of a bounded number of positive nth powers, and Goldbach’s Conjecture, which asserts that every positive even number greater than 2 is the sum of two primes. The book 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”.

Waring’s Problem was solved in 1909 by David Hilbert, although with a pure existence proof that did not give any idea of the number of powers required. This book presents a complete exposition of Hilbert’s proof as simplified by several later authors. About ten years after Hilbert's work, G. H. Hardy and J. E. Littlewood used the “circle method” (originally developed by Hardy and Ramanujan for work on partitions) to obtain asymptotic estimates for the number of representations. This book develops the circle method in general and uses it to develop these estimates.

Goldbach’s Conjecture is still unsolved, although much progress has been made and this book proves the main results. I. M. Vinogradov developed the method of trigonometrical sums, based on some of the same ideas used in the circle method, and used trigonometrical sums in 1937 to prove that every sufficiently large odd number is the sum of three primes. The twin prime problem (that asks whether there are infinitely many primes p such that p+2 is also prime) has driven most of the work on sieve methods in the twentieth century, and much of this work can be applied with modest changes to the Goldbach Conjecture. Using sieve methods, J.-R. Chen proved in 1965 that every sufficiently large even number is the sum of a prime and a number that is either prime or the product of two primes. The present book presents proofs of Vinogradov’s and Chen’s theorems.

The book works up to the main problems with a series of simpler, historically-important problems. For Waring’s problem it deals first with sums of squares, of cubes, and of polygonal numbers. For the Goldbach Conjecture it first develops Brun’s 1920 upper estimate on the number of twin primes and proves Shnirelman’s 1930 theorem that every positive integer is the sum of a bounded number of primes. The book also includes a large number of exercises, but most of these are much simpler than the problems considered in the body of the text.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.


  • Preface
  • Notation and conventions
  • I. Waring’s problem
    1. Sums of polygons
      1. Polygonal numbers
      2. Lagrange’s theorem
      3. Quadratic forms
      4. Ternary quadratic forms
      5. Sums of three squares
      6. Thin sets of squares
      7. The polygonal number theorem
      8. Notes
      9. Exercises
    2. Waring’s problem for cubes
      1. Sums of cubes
      2. The Wieferich-Kempner theorem
      3. Linnik’s theorem
      4. Sums of two cubes
      5. Notes
      6. Exercises
    3. The Hilbert-Waring theorem
      1. Polynomial identities and a conjecture of Hurwitz
      2. Hermite polynomials and Hilbert’s identity
      3. A proof by induction
      4. Notes
      5. Exercises
    4. Weyl’s inequality
      1. Tools
      2. Difference operators
      3. Easier Waring’s problem
      4. Fractional parts
      5. Weyl’s inequality and Hua’s lemma
      6. Notes
      7. Exercises
    5. The Hardy-Littlewood asymptotic formula
      1. The circle method
      2. Waring’s problem for k = 1
      3. The Hardy-Littlewood decomposition
      4. The minor arcs
      5. The major arcs
      6. The singular integral
      7. The singular series
      8. Conclusion
      9. Notes
      10. Exercises
  • II. The Goldbach conjecture
    1. Elementary estimates for primes
      1. Euclid’s theorem
      2. Chebyshev’s theorem
      3. Mertens’s theorems
      4. Brun’s method and twin primes
      5. Notes
      6. Exercises
    2. The Shnirel’man-Goldbach theorem
      1. The Goldbach conjecture
      2. The Selberg sieve
      3. Applications of the sieve
      4. Shnirel’man density
      5. The Shnirel’man-Goldbach theorem
      6. Romanov’s theorem
      7. Covering congruences
      8. Notes
      9. Exercises
    3. Sums of three primes
      1. Vinogradov’s theorem
      2. The singular series
      3. Decomposition into major and minor arcs
      4. The integral over the major arcs
      5. An exponential sum over primes
      6. Proof of the asymptotic formula
      7. Notes
      8. Exercise
    4. The linear sieve
      1. A general sieve
      2. Construction of a combinatorial sieve
      3. Approximations
      4. The Jurkat-Richert theorem
      5. Differential-difference equations
      6. Notes
      7. Exercises
    5. Chen’s theorem
      1. Primes and almost primes
      2. Weights
      3. Prolegomena to sieving
      4. A lower bound for S(A, P, z)
      5. An upper bound for S(Aq, P, z)
      6. An upper bound for S(B, p, y)
      7. A bilinear form inequality
      8. Conclusion
      9. Notes
  • III. Appendix: Arithmetic functions
    1. The ring of arithmetic functions
    2. Sums and integrals
    3. Multiplicative functions
    4. The divisor function
    5. The Euler φ-function
    6. The Möbius function
    7. Ramanujan sums
    8. Infinite products
    9. Notes
    10. Exercises
  • Bibliography
  • Index