## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  146.27201
Autor:  Erdös, Pál
Title:  Remarks on number theory. V: Extremal problems in number theory. II (In Hungarian. English summary)
Source:  Mat. Lapok 17, 135-155 (1966).
Review:  The author continues the investigation of extremal problems in number theory started in another paper (this Zbl 127.02202). In this summary I just state a few of the problems and results considerd. Let a1 < ··· < a2 \leq n be a sequence of integers such that the products prodi = 1z ai\epsiloni, \epsiloni = 0 or 1, are all different. Then z < \pi (n)+c1n ½/ log n (this was conjectured in loc. cit.) Perhaps z < \pi(n)+\pi(n ½)+o(n ½/ log n). Let a1 < a2 < ··· be an infinite sequence of integers for which the sums ai+aj are all distinct. Can one have ak = o(k3)? Let a1 < a2 ··· be an infinite sequence of integers no a divides any other. Sárközi, Szeméredi and I proved (sharpening a previous result of Behrend) that sumai < x 1/ai = o(log x/ log log x) ½).
Classif.:  * 11B75 Combinatorial number theory
11B83 Special sequences of integers and polynomials
Index Words:  number theory

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