Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  518.10063
Autor:  Erdös, Paul; Freud, R.; Hegyvari, N.
Title:  Arithmetical properties of permutations of integers. (In English)
Source:  Acta Math. Hung. 41, 169-176 (1983).
Review:  Let a1,...,an be a permutation of 1,...,n and let [ai,aj] denote the least common multiple of ai and aj. It is shown that

maxmax1 \leq i < n[ai,ai+1] = (1+o(1))\frac{n2}{4 log n},

where the minimum is taken over all permutations. This result is best possible since in any permutation there must be an ai such that [ai,ai+1] \geq (1+o(1))\frac{n2}{4 log n}. It is also shown that there is an infinite permutation a1,a2,... of the positive integers such that

[ai,ai+1] < iec\sqrt{log i log log i}

for all i. Some results are also obtained for the greatest common divisor. See also following review.
Reviewer:  I.Anderson
Classif.:  * 11B05 Topology etc. of sets of numbers
                   11A05 Multiplicative structure of the integers
                   11B75 Combinatorial number theory
                   05A05 Combinatorial choice problems
Keywords:  permutations; density of sums; least common multiple; greatest common divisor
Citations:  Zbl.518.10064

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