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**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 a_{1} < ··· < a_{2} \leq n be a sequence of integers such that the products **prod**_{i = 1}^{z} a_{i}^{\epsiloni}, \epsilon_{i} = 0 or 1, are all different. Then z < \pi (n)+c_{1}n^{ ½}/ log n (this was conjectured in loc. cit.) Perhaps z < \pi(n)+\pi(n^{ ½})+o(n^{ ½}/ log n). Let a_{1} < a_{2} < ··· be an infinite sequence of integers for which the sums a_{i}+a_{j} are all distinct. Can one have a_{k} = o(k^{3})? Let a_{1} < a_{2} ··· be an infinite sequence of integers no a divides any other. Sárközi, Szeméredi and I proved (sharpening a previous result of Behrend) that **sum**_{ai < x} 1/a_{i} = 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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