Difference between revisions of "Shnirel'man method"
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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 method was applied to prove that the sequence consisting of the number one and of the numbers of the form | + | 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==== | ||
Revision as of 08:13, 6 June 2020
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) |
Shnirel'man method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shnirel%27man_method&oldid=48688