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**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 n_{1},...,n_{f}, 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 n_{1},...,n_{f} is a perfect power (**prod**_{i = 1}^{f}n_{i} in **N**^{m} for som m \geq 2). (2) Two distinct subsets of **{**n_{1},...,n_{f}**}** yield the same product (**prod**_{i in I1} n_{1} = **prod**_{i in I2} n_{i}). (3) n_{1},...,n_{f} are multiplicatively dependent (**prod**_{i in I1} n_{1}^{mi} = **prod**_{i in I2} n_{i}^{mi} for certain m_{i} in **N**). (4) The total number of distinct primes occurring in the prime factorizations of the integers n_{1},...,n_{f} is less than the number integers (\omega(**prod**_{i = 1}^{f}n_{i}) < 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+c_{1}(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(c_{2}(log n log log n)^{ ½ })], but for infinitely many n in **N** there exists two distinct sets of integers in [n,n+\exp(c_{3}(log n log log n)^{ ½ })], with equal products and for all n in **N** the latter holds for [n,n+c_{4}n^{0.496}]. The c_{1},c_{2},c_{3},c_{4} 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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