Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 497.10033
Autor: Erdös, Paul; Turk, J.
Title: Products of integers in short intervals. (In English)
Source: Acta Arith. 44, 147-174 (1984).
Review: The following properties of distinct integers, say n1,...,nf, from a ``short'' interval [n,n+k(n)], where k(n) is a ``small'' function of n (such as n ½ , or log n) and n \geq 1 is arbitrary, are considered: (1) The product of n1,...,nf is a perfect power (prodi = 1fni in Nm for som m \geq 2). (2) Two distinct subsets of {n1,...,nf} yield the same product (prodi in I1 n1 = prodi in I2 ni). (3) n1,...,nf are multiplicatively dependent (prodi in I1 n1mi = prodi in I2 nimi for certain mi in N). (4) The total number of distinct primes occurring in the prime factorizations of the integers n1,...,nf is less than the number integers (\omega(prodi = 1fni) < f). Our results can be summarized as follows: the above properties never occur in ``very short'' intervals, sometimes in ``short'' intervals and always in ``large'' intervals. For example, distinct sets of integers from [n,n+c1(log n)2(log log n)-1] have distinct products for any n \geq 3, for infinitely many n in N this also holds for [n,n+\exp(c2(log n log log n) ½ )], but for infinitely many n in N there exists two distinct sets of integers in [n,n+\exp(c3(log n log log n) ½ )], with equal products and for all n in N the latter holds for [n,n+c4n0.496]. The c1,c2,c3,c4 are absolute positive constants.
Reviewer: P.Erdös
Classif.: * 11N05 Distribution of primes
11D41 Higher degree diophantine equations
11D61 Exponential diophantine equations
Keywords: distinct sets of integers; distinct products; equal products; consecutive
integers; integers in short intervals; products of integers; perfect powers
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