1. Introduction. For any integer n > 1 let P(n) denote the greatest prime factor of n. Győry, Sárközy and Stewart [5] conjectured that if a, b and c are pairwise distinct positive integers then (1) P((ab+1)(bc+1)(ca+1)) tends to infinity as max(a,b,c) → ∞. In this paper we confirm this conjecture in the special case when at least one of the numbers a, b, c, a/b, b/c, c/a has bounded prime factors. We prove our result in a quantitative form by showing that if is a finite set of triples (a,b,c) of positive integers a, b, c with the property mentioned above then for some (a,b,c) ∈ , (1) is greater than a constant times log||loglog||, where || denotes the cardinality of (cf. Corollary to Theorem 1). Further, we show that this bound cannot be replaced by (cf. Theorem 2). Recently, Stewart and Tijdeman [9] proved the conjecture in another special case. Namely, they showed that if a ≥ b > c then (1) exceeds a constant times log((loga)/log(c+1)). In the present paper we give an estimate from the opposite side in terms of a (cf. Theorem 3).
@article{bwmeta1.element.bwnjournal-article-aav79i2p163bwm,
author = {K. Gy\H ory and A. S\'ark\"ozy},
title = {On prime factors of integers of the form (ab+1)(bc+1)(ca+1)},
journal = {Acta Arithmetica},
volume = {80},
year = {1997},
pages = {163-171},
zbl = {0869.11071},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav79i2p163bwm}
}
K. Győry; A. Sárközy. On prime factors of integers of the form (ab+1)(bc+1)(ca+1). Acta Arithmetica, Tome 80 (1997) pp. 163-171. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav79i2p163bwm/
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