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**Zentralblatt MATH**

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

**Zbl.No: ** 151.03502

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

**Title: ** On the solvability of the equations [a_{i},a_{j}] = a_{r} and (a'_{i},a'_{j}) = a'_{r} in sequences of positive density (In English)

**Source: ** J. Math. Anal. Appl. 15, 60-64 (1966).

**Review: ** The authors obtain the following results.

1) Let a_{1} < a_{2} < ··· be an infinite sequence of integers for which there are infinitely many integers n_{1} < n_{2} < ··· satisfying **sum**_{ai < nk} {1 \over a_{i}} > c_{1} {log n_{k} \over (log log n_{k})^{ ½}}. Then the equations (a'_{i},a'_{j}) = a'_{r},[a_{i},a_{j}] = a_{r} have infinitely many solutions. The symbol (a_{i},a_{j}) denotes the greatest common divisor and [a_{i},a_{j}] denotes the least common multiple of a_{i} and a_{j}.

2) Let a_{1} < a_{2} < ··· be an infinite sequence of integers for which there are infinitely many integers n_{1} < n_{2} < ··· satisfying

**sum**_{ai < nk} {1 \over a_{i}} > c_{2} {log n_{k} \over (log log n_{k})^{1/4}}. Then there are infinitely many quadruplets of distincts integers a_{i},a_{j},a_{r},a_{s} satisfying (a_{i},a_{j}) = a_{r}, [a_{i},a_{j}] = a_{s}, c_{1} and c_{2} denote suitable positive constants.

**Reviewer: ** Cs.Pogany

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

**Index Words: ** number theory

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