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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Divisibility properties of additive bases


Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 96 (1986), 11-14
MSC: Primary 11B99; Secondary 11P99
MathSciNet review: 813799
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Abstract: Let $ h \geqslant 2$. There exists an asymptotic basis $ A$ of order $ h$ such that $ ({a_1}, \ldots ,{a_k}){\text{ > 1}}$ for all $ {a_1} \ldots ,{a_k} \in A$ if and only if $ k{\text{ < }}h$. If $ k \geqslant h$, the sumset $ hA$ contains only composite numbers. For $ h = k$, there exists a set $ A$ of nonnegative integers with $ ({a_1}, \ldots ,{a_h}){\text{ > }}1$ for all $ {a_1}, \ldots ,{a_h} \in A$ such that for every prime $ p$ the sumset $ hA$ contains all sufficiently large multiples of $ p$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0813799-X
PII: S 0002-9939(1986)0813799-X
Keywords: Density of sumsets, asymptotic bases, additive number theory, additive bases
Article copyright: © Copyright 1986 American Mathematical Society