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

• Autorzy: K. Győry , A. Sárközy
• Języki publikacji: EN

Abstrakty

EN

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 𝓐 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) ∈ 𝓐, (1) is greater than a constant times log|𝓐|loglog|𝓐|, where |𝓐| denotes the cardinality of 𝓐 (cf. Corollary to Theorem 1). Further, we show that this bound cannot be replaced by $|𝓐|^ε$ (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).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1997
• Tom: 79
• Numer: 2
• Strony: 163-171

Daty

wydano

1997

otrzymano

1996-09-06

Twórcy

autor

K. Győry

• Institute of Mathematics, Kossuth Lajos University, 4010 Debrecen, Hungary

autor

A. Sárközy

• Mathematical Institute, Hungarian Academy of Sciences, 1053 Budapest, Hungary

Bibliografia

• [1] A. Balog and I. Z. Ruzsa, On an additive property of stable sets, in: Proc. Cardiff Number Theory Conf., 1995, to appear.
• [2] P. Erdős, C. L. Stewart and R. Tijdeman, Some diophantine equations with many solutions, Compositio Math. 66 (1988), 37-56.
• [3] J. H. Evertse, On equations in S-units and the Thue-Mahler equation, Invent. Math. 75 (1984), 561-584.
• [4] J. H. Evertse, The number of solutions of decomposable form equations, Invent. Math. 122 (1995), 559-601.
• [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.
• [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.
• [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.