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**Zentralblatt MATH**

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

**Zbl.No: ** 476.10045

**Autor: ** Erdös, Pál; Szemeredi, Endre

**Title: ** Remarks on a problem of the American Mathematical Monthly. (In Hungarian)

**Source: ** Mat. Lapok 28, 121-124 (1980).

**Review: ** Let A = a_{1} < a_{2} < ... be a sequence of positive integers. Let F(A,x,i) denote the number of k's for which the least common multiple [a_{k},a_{k+1},...,a_{k+i-1}] satisfies in the inequality [a_{k},a_{k+1},...,a_{k+i-1}] \leq x. Some years ago *P.Erdös* formulated the problem in Am. Math. Mon. to prove that F(A,x,i) < c_{i} x^{1/i}, where c_{i} is a constant depending only on i. This statement is false. This can be seen from the following results of the paper (see III).

I. For any A we have **lim**\sup_{x ––> oo} \frac{F(A,x,2)}{\sqrt{x}} \leq **sum**_{k = 1}^{oo} \frac{k^{ ½}-(k-1)^{ ½}}{k} and

**liminf**_{x ––> oo} \frac{F(A,x,2)}{\sqrt{x}} = 0 (1) provided that in (1) the sign = holds.

II. For any A we have **liminf**_{x ––> oo} \frac{F(A,x,2)}{\sqrt{x}} \leq ½ .

III. If i > 4, then there exists an \alpha_{i} > 0 such that for each sufficiently large x and suitable A we have

F(A,x,i) > x^{ 1/i +\alphai}. The authors conjecture that III holds for i = 4, too. For i = 3 they have proved that for each sufficiently large x and any A F(A,x,3) < c_{0}x^{1/3} log x (c_{0} > 0) and that there is such an A that F(A,x,3) > c_{1}x^{1/3} log x (c_{1} > 0) for infinitely many x holds. The question whether there is such an A that for each x the inequality F(A,x,3) > c_{2}x^{1/3} log x (c_{2}) holds, remain open.

**Reviewer: ** T.Salát

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

11A05 Multiplicative structure of the integers

**Keywords: ** sequence of positive integers; least common multiple

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