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On a Problem of P. Erdös

Published online by Cambridge University Press:  20 November 2018

I. Ruzsa Jr.
I. Ruzsa Jr.*
Affiliation:
Fazekas High School, Budapest, Hungary
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P. Erdös asked the following problem: Does there exist an infinite sequence of integers a1<…satisfying for every x≥1

1

so that every integer is of the form 2k+ai [1]. The analogous questions can easily be answered affirmatively if the powers of 2 are replaced by the rth power.

In this note we give a simple affirmative answer to the problem of Erdôs. Let c2 be a sufficiently small absolute constant. Our sequence A consists of all the integers of the form

2

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 309 - 310
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Erdös, P., Some results on additive number theory, Proc. Amer. Math. Soc. 5 (1954), 847-853 (see p. 853). See also Proc. of the Number Theory Conf. at Boulder, Colorado, 1963, Problem 33.Google Scholar
2. Lorentz, G. G., On a problem of additive number theory, Proc. Amer. Math. Soc. 5 (1954), 838-891.Google Scholar
3. Moser, L., On the additive completion of sets of integers, Proc. Symp. Pure Math., Amer. Math. Soc. 8 (1965), 175-180.Google Scholar