# 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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topK. 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

top- [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] P. Erdős, C. L. Stewart and R. Tijdeman, Some diophantine equations with many solutions, Compositio Math. 66 (1988), 37-56. Zbl0639.10014
- [3] J. H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561-584. Zbl0521.10015
- [4] J. H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), 559-601. Zbl0851.11019
- [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] A. Hildebrand, On a conjecture of Balog, Proc. Amer. Math. Soc. 95 (1985), 517-523.
- [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] T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge Univ. Press, 1986.
- [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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