Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 869.11001
Autor: Erdös, Paul; Evans, Anthony B.
Title: Sets of prime numbers satisfying a divisibility condition. (In English)
Source: J. Number Theory 61, No.1, 39-43, Art. No.0135 (1996).
Review: Let P be a set of prime numbers. For any subset A of P let \Pi A denote the product of all primes in A. The set P is said to satisfy condition (*) if {gcd} (\Pi A-\Pi B, \Pi P) = 1 for all disjoint, non-empty subsets A,B of P. The authors have previously proved [J. Graph Theory 13, No. 5, 593-595 (1989; Zbl 691.05053)] that for all k there exists a set P of k primes satisfying (*). Now let nk be the smallest \Pi P, where P is a set of k primes satisfying (*).
Theorem 1: If P is a set of k primes, k \geq 2, satisfying (*), and p is the smallest prime in P, then k \leq log2 (p-1)+2. Further, if P cannot be extended to a set of k+1 primes satisfying (*) then
k \geq {Min} (r: 3r-1-2r-1 \geq p-1) = {one of} \lceil log3(p-1) \rceil+1 {or} \lceil log2(p-1) \rceil+2. Theorem 2: (a) For k \geq 2,
(log2nk)/k2 > 1-2/k. (b) For \epsilon > 0,
(log2 nk)/k2 < log2 (3+\epsilon) for all k sufficiently large.
Reviewer: B.Garrison (San Diego)
Classif.: * 11A05 Multiplicative structure of the integers
Keywords: products of primes; greatest common divisor; set of prime numbers
Citations: Zbl 691.05053
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