Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  401.10068
Autor:  Erdös, Paul; Sarközy, A.
Title:  On products of integers. II. (In English)
Source:  Acta Sci. Math. 40, 243-259 (1978).
Review:  Let k,n be any positive integers. A = {a1,...,an} any finite, strictly increasing sequence of positive integers satisfying (*) a1 = 1, a2 = 2, ... ak = k. Let us denote the number of integers which can be written in the form prodi = 1nai\epsiloni(\epsilon1 = 0 or 1) or aiaj (1 \leq i, j \leq n), respectively by f(A,n,k) and g(A,n,k). Let us write F(n,k) = maxAf(A,n,k) and G(n,k) = maxAg(A,n,k), where the minima are extended over all sequence A satisfying (*) and |A| = n. The authors conjectured in an earlier paper [Studia Sci. Math. Hung. 9, 161-171 (1974; Zbl 304.10034)] that (1) G(n,k)/n > c1. G(k,k)/k for every n \geq k, and furthermore, that for any \omega > o, k > k0(\omega) and n \geq k, we have F(n,k) > n2k\omega or perhaps (2) n2\exp(\frac{c2k}{log k}) < F(n,k) < n2\exp(c3k/ log k) for large k and n \geq k. In this paper, the authors disprove (1) and prove a slightly weaker form of (2).
Reviewer:  D.Suryanarayana
Classif.:  * 11B83 Special sequences of integers and polynomials
                   11N13 Primes in progressions
Keywords:  products of integers; distribution in sequences
Citations:  Zbl.304.10034

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