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)