## 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 [ai,aj] = ar 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 a1 < a2 < ··· be an infinite sequence of integers for which there are infinitely many integers n1 < n2 < ··· satisfying

sumai < nk {1 \over ai} > c1 {log nk \over (log log nk) ½}.

Then the equations (a'i,a'j) = a'r,[ai,aj] = ar have infinitely many solutions. The symbol (ai,aj) denotes the greatest common divisor and [ai,aj] denotes the least common multiple of ai and aj.
2) Let a1 < a2 < ··· be an infinite sequence of integers for which there are infinitely many integers n1 < n2 < ··· satisfying

sumai < nk {1 \over ai} > c2 {log nk \over (log log nk)1/4}.

Then there are infinitely many quadruplets of distincts integers ai,aj,ar,as satisfying (ai,aj) = ar, [ai,aj] = as, c1 and c2 denote suitable positive constants.
Reviewer:  Cs.Pogany
Classif.:  * 11B83 Special sequences of integers and polynomials
Index Words:  number theory

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