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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 159.06003

**Autor: ** Erdös, Pál; Sarközi, A.; Szemeredi, E.

**Title: ** On the solvability of some equations in dense sequences of integers (In English. RU original)

**Source: ** Sov. Math., Dokl. 8, 1160-1164 (1967); translation from Dokl. Akad. Nauk SSSR 176, 541-544 (1967).

**Review: ** Let a_{1} < a_{2} < ... be an infinite sequence of integers for which the equation (a_{i},a_{j}) = a_{r} is unsolvable for any set of distinct indices i,j,r. The authors prove that then **sum**_{k = 1}^{oo} {1 \over a_{k} log a_{k}} < oo. (1) The proof is elementary but quite complicated and uses combinatorial arguments. In a previous paper [*P. Erdös*, J. London Math. Soc. 10, 126-128 (1935; Zbl 012.05202)] the following weaker result was proved: Assume that no a_{i} divides any other than (1) holds. In another paper [J. Math. Anal. Appl. 15, 60-64 (1966; Zbl 151.03502)] the authors point out that (1) does not hold if we assume that [a_{i},a_{j}] = a_{r} is unsolvable for any set of distinct indices i,j,r.

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11B75 Combinatorial number theory

**Index Words: ** number theory

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