On prime factors of integers of the form (ab+1)(bc+1)(ca+1)

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 ${\displaystyle \mid {\mid }^{\epsilon }}$ (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).