Additive problems

2020 Mathematics Subject Classification: Primary: 11P [MSN][ZBL]

Problems in number theory concerning the decomposition (or partition) of integers into summands of a given kind. The solution of classical additive problems resulted in the development of new methods in number theory. The classical additive problems include:

1) The Goldbach problem on representing an odd integer, larger than 5, as the sum of three prime numbers, and the Euler–Goldbach problem on the representation of an even integer, larger than 2, as the sum of two prime numbers. These problems were first posed in 1742.

2) The Waring problem on representing any positive integer as a sum of $s=s(k)$ non-negative $k$-th powers, where $k\geq2$ is given. First posed in 1770.

Other additive problems are, for example:

3) The problem on the representation of positive integers as the sum of a bounded number of prime numbers (the weak Goldbach problem).

4) The Hardy–Littlewood problem on representing any integer, larger than 1, as the sum of a prime number and two squares (posed in the 1920s).

5) Problems concerning the representation of all sufficiently large numbers as sums of two numbers with a bounded number of prime factors.

6) Problems concerning the representation of integers by quadratic forms in three and four variables, and similar problems.

Additive problems may be solved by analytic, algebraic, elementary and mixed methods (cf. Additive number theory). A large number of additive problems belong to one of the following two classes:

a) Ternary additive problems of the type $n=\alpha+\beta+\gamma$, where $\alpha$ and $\beta$ belong to sufficiently dense sequences of integers which are well distributed in arithmetic progressions, while $\gamma$ belongs to a sequence which may be thin, but for which the corresponding relevant trigonometric sums behave properly.

b) Binary additive problems of the type $n=\alpha+\beta$, with $\alpha$ and $\beta$ subject to the same conditions as those stated in a).

A universal tool for solving ternary additive problems for sufficiently large $n$ is the general analytic method of Hardy–Littlewood–Vinogradov: the method of trigonometric sums (cf. Vinogradov method). Binary additive problems usually cannot be solved by such methods, and are solved by different variants of the elementary sieve method (cf. Sieve method). Especially powerful results are obtained by the method of the large sieve and by the dispersion method, due to Yu.V. Linnik. Additive problems of the type 6) above are also binary. They are studied by special arithmetic-geometric methods of the theory of quadratic forms.

References

[1]	I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) MR0603100 MR0409380
[2]	Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian) MR0168543 Zbl 0112.27402
[3]	Yu.V. Linnik, "Ergodic properties of algebraic fields" , Springer (1968) (Translated from Russian) MR0238801 Zbl 0162.06801
[4]	L.-K. Hua, "Abschätzungen von Exponentialsummen und ihre Anwendung in der Zahlentheorie" , Enzyklopaedie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen , 1 : 2 (1959) (Heft 13, Teil 1) Zbl 0083.03601