##
**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 a_{1},...,a_{n} be a permutation of 1,...,n and let [a_{i},a_{j}] denote the least common multiple of a_{i} and a_{j}. It is shown that **max****max**_{1 \leq i < n}[a_{i},a_{i+1}] = (1+o(1))\frac{n^{2}}{4 log n}, where the minimum is taken over all permutations. This result is best possible since in any permutation there must be an a_{i} such that [a_{i},a_{i+1}] \geq (1+o(1))\frac{n^{2}}{4 log n}. It is also shown that there is an infinite permutation a_{1},a_{2},... of the positive integers such that

[a_{i},a_{i+1}] < ie^{c\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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag