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Difference between revisions of "Asymptotic basis"

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''asymptotic basis of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013620/a0136202.png" />''
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''asymptotic basis of order $k$''
  
A sequence of natural numbers and zero, which as a result of its summation repeated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013620/a0136203.png" /> times yields all sufficiently large natural numbers. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013620/a0136204.png" /> is called the order of the asymptotic basis. Thus, the sequence of prime numbers is an asymptotic basis of order 4 (I.M. Vinogradov, 1937); the sequence of cubes of natural numbers is an asymptotic basis of order 7 (Yu.V. Linnik, 1942).
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A sequence of natural numbers and zero, which as a result of its summation repeated $k$ times yields all sufficiently large natural numbers. The number $k$ is called the order of the asymptotic basis. Thus, the sequence of prime numbers is an asymptotic basis of order 4 (I.M. Vinogradov, 1937); the sequence of cubes of natural numbers is an asymptotic basis of order 7 (Yu.V. Linnik, 1942).
  
  

Latest revision as of 16:33, 15 April 2014

asymptotic basis of order $k$

A sequence of natural numbers and zero, which as a result of its summation repeated $k$ times yields all sufficiently large natural numbers. The number $k$ is called the order of the asymptotic basis. Thus, the sequence of prime numbers is an asymptotic basis of order 4 (I.M. Vinogradov, 1937); the sequence of cubes of natural numbers is an asymptotic basis of order 7 (Yu.V. Linnik, 1942).


Comments

See also Waring problem; Goldbach problem.

How to Cite This Entry:
Asymptotic basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_basis&oldid=31758
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article