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

K. Győry, and A. Sárközy. "On prime factors of integers of the form (ab+1)(bc+1)(ca+1)." Acta Arithmetica 79.2 (1997): 163-171. <http://eudml.org/doc/206973>.

@article{K1997,
abstract = {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 $||^ε$ (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).},
author = {K. Győry, A. Sárközy},
journal = {Acta Arithmetica},
keywords = {greatest prime factor},
language = {eng},
number = {2},
pages = {163-171},
title = {On prime factors of integers of the form (ab+1)(bc+1)(ca+1)},
url = {http://eudml.org/doc/206973},
volume = {79},
year = {1997},
}

TY - JOUR
AU - K. Győry
AU - A. Sárközy
TI - On prime factors of integers of the form (ab+1)(bc+1)(ca+1)
JO - Acta Arithmetica
PY - 1997
VL - 79
IS - 2
SP - 163
EP - 171
AB - 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 $||^ε$ (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).
LA - eng
KW - greatest prime factor
UR - http://eudml.org/doc/206973
ER -