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Primitive polynomial subsequences

Published online by Cambridge University Press:  26 February 2010

I. Anderson ,
S. D. Cohen  and
W. W. Stothers
I. Anderson
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
S. D. Cohen
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
W. W. Stothers
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW
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Extract

A sequence {an} of integers is said to be primitive if whenever i ≠ j. For example, if n is any positive integer, the sequence

is primitive. This is an important example in the light of the elementary result (see [2; p. 244]) that if 0 < a1 < a2 < … < ar ≤ 2n is primitive then necessarily r ≤ n; i.e. at most half of the positive integers ≤ 2n can be members of the sequence. Besicovitch [2; p. 257] has obtained the surprising result that, given ε > 0, there exists an infinite primitive sequence {ai} such that

where A(n) denotes the number of ai ≤ n.

Type
Research Article
Information
Mathematika , Volume 21 , Issue 2 , December 1974 , pp. 239 - 247
Copyright
Copyright © University College London 1974

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References

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