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A method for adding sequences of positive integers; created by L.G. Shnirel'man in 1930. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849301.png" /> be the number of elements of the sequence not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849302.png" />. Similarly to the measure of a set, one defines
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849303.png" /></td> </tr></table>
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the density of the sequence. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849304.png" /> the elements of which are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849307.png" />, is called the sum of the two sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s0849309.png" />.
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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
  
Shnirel'man's theorem 1): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493010.png" /> are the densities of the summands, then the density of the sum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493011.png" />. 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.
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$$
 +
\alpha  = \inf _ {n = 1,2,\dots } 
 +
\frac{\nu ( n) }{n}
 +
,
 +
$$
  
Shnirel'man's theorem 2): The sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493012.png" /> has positive density, where the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493013.png" /> consists of the number one and all prime numbers; hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493014.png" /> is a basis of the natural series, i.e. every natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493015.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493016.png" /> has been obtained. In the case of representing a sufficiently large number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493017.png" /> by a sum of prime numbers with number of summands <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493018.png" /> (Shnirel'man's constant), Shnirel'man's method together with analytical methods gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493019.png" />. However, by the more powerful method of trigonometric sums of I.M. Vinogradov (cf. [[Trigonometric sums, method of|Trigonometric sums, method of]]) the estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493020.png" /> was obtained.
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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 method was applied to prove that the sequence consisting of the number one and of the numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493022.png" /> is a prime number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493023.png" /> is a natural number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084930/s08493024.png" /> is a basis of the natural series (N.P. Romanov, 1934).
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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====
 
====References====
 
<table>
 
<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">[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) 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>
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<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>
 
<TR><TD valign="top">[3]</TD> <TD valign="top">  K. Prachar,  "Primzahlverteilung" , Springer  (1957)</TD></TR>
 
</table>
 
</table>

Latest revision as of 19:05, 20 November 2023


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)
How to Cite This Entry:
Shnirel'man method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shnirel%27man_method&oldid=36152
This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article