Shnirel'man method

A method for adding sequences of positive integers; created by L.G. Shnirel'man in 1930. Let $\nu ( x) \neq 0$ be the number of elements of the sequence not larger than $x$. Similarly to the measure of a set, one defines

$$\alpha = \inf _ {n = 1,2,\dots } \frac{\nu ( n) }{n} ,$$

the density of the sequence. A sequence $C$ the elements of which are $c = a+ b$, $a \in A$, $b \in B$, is called the sum of the two sequences $A$ and $B$.

Shnirel'man's theorem 1): If $\alpha , \beta$ are the densities of the summands, then the density of the sum is $\gamma = \alpha + \beta - \alpha \beta$. If after adding a sequence to itself a finite number of times one obtains the entire natural series, then the initial sequence is called a basis. In this case every natural number can be represented as the sum of a limited number of summands of the given sequence. A sequence with positive density is a basis.

Shnirel'man's theorem 2): The sequence ${\mathcal P} + {\mathcal P}$ has positive density, where the sequence ${\mathcal P}$ consists of the number one and all prime numbers; hence, ${\mathcal P}$ is a basis of the natural series, i.e. every natural number $n \geq 2$ can be represented as the sum of a limited number of prime numbers. For the number of summands (Shnirel'man's absolute constant) the estimate $S \leq 19$ has been obtained. In the case of representing a sufficiently large number $n \geq n _ {0}$ by a sum of prime numbers with number of summands $S$( Shnirel'man's constant), Shnirel'man's method together with analytical methods gives $S \leq 6$. However, by the more powerful method of trigonometric sums of I.M. Vinogradov (cf. Trigonometric sums, method of) the estimate $S \leq 4$ was obtained.

Shnirel'man's method was applied to prove that the sequence consisting of the number one and of the numbers of the form $p + a ^ {m}$, where $p$ is a prime number, $a \geq 2$ is a natural number and $m = 1, 2 \dots$ is a basis of the natural series (N.P. Romanov, 1934).

References

[1]	L.G. [L.G. Shnirel'man] Schnirelmann, "Ueber additive Eigenschaften von Zahlen" Math. Ann. , 107 (1933) pp. 649–690
[2]	A.Ya. Khinchin, "Three pearls of number theory" , Graylock (1952) Translation from the second, revised Russian ed. [1948] Zbl 0048.27202 Reprinted Dover (2003) ISBN 0486400263
[3]	K. Prachar, "Primzahlverteilung" , Springer (1957)