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

Acta Arithmetica (1997)

• Volume: 79, Issue: 2, page 163-171
• ISSN: 0065-1036

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## Abstract

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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 ${||}^{\epsilon }$ (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).

## How to cite

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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 -

## References

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1. [1] A. Balog and I. Z. Ruzsa, On an additive property of stable sets, in: Proc. Cardiff Number Theory Conf., 1995, to appear. Zbl0924.11011
2. [2] P. Erdős, C. L. Stewart and R. Tijdeman, Some diophantine equations with many solutions, Compositio Math. 66 (1988), 37-56. Zbl0639.10014
3. [3] J. H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561-584. Zbl0521.10015
4. [4] J. H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), 559-601. Zbl0851.11019
5. [5] K. Győry, A. Sárközy and C. L. Stewart, On the number of prime factors of integers of the form ab+1, Acta Arith. 74 (1996), 365-385. Zbl0857.11047
6. [6] A. Hildebrand, On a conjecture of Balog, Proc. Amer. Math. Soc. 95 (1985), 517-523.
7. [7] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. Zbl0122.05001
8. [8] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Univ. Press, 1986.
9. [9] C. L. Stewart and R. Tijdeman, On the greatest prime factor of (ab+1)(ac+1)(bc+1), this volume, 93-101. Zbl0869.11072

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