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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|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  L.G. [L.G. Shnirel'man] Schnirelmann,   "Ueber additive Eigenschaften von Zahlen"  ''Math. Ann.'' , '''107'''  (1933)  pp. 649–690</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  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}}</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,   "Primzahlverteilung" , Springer  (1957)</TD></TR>
</table>