Corvaja, P.; Zannier, U. On the greatest prime factor of \((ab+1)(ac+1)\). (English) Zbl 1077.11052 Proc. Am. Math. Soc. 131, No. 6, 1705-1709 (2003). The method introduced by the authors in [Indag. Math. 9, 317–332 (1998; Zbl 0923.11103)] (which applies the subspace theorem) is used to prove that the greatest prime factor of \((ab+1)(ac+1)\) tends to infinity with \(a\). This solves a long standing open problem for which earlier partial results and conjectures were obtained e.g. by K. Győry, A. Sárközy and C. L. Stewart [Acta Arith. 74, 365–385 (1996; Zbl 0857.11047)]. Reviewer: Istvan Gaál (Debrecen) Cited in 4 ReviewsCited in 20 Documents MSC: 11J25 Diophantine inequalities Keywords:greatest prime factor; diophantine inequalities Citations:Zbl 0923.11103; Zbl 0857.11047 PDF BibTeX XML Cite \textit{P. Corvaja} and \textit{U. Zannier}, Proc. Am. Math. Soc. 131, No. 6, 1705--1709 (2003; Zbl 1077.11052) Full Text: DOI OpenURL Online Encyclopedia of Integer Sequences: 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0. 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0. 7-smooth but not 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0. a(n) is the greatest prime factor of A180045(n). References: [1] Yann Bugeaud, On the greatest prime factor of (\?\?+1)(\?\?+1)(\?\?+1), Acta Arith. 86 (1998), no. 1, 45 – 49. · Zbl 0941.11014 [2] Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of \(a^{n}-1\) and \(b^{n}-1\), to appear in Math. Zeit. · Zbl 1021.11001 [3] P. Corvaja and U. Zannier, Diophantine equations with power sums and universal Hilbert sets, Indag. Math. (N.S.) 9 (1998), no. 3, 317 – 332. · Zbl 0923.11103 [4] K. Győry and A. Sárközy, On prime factors of integers of the form (\?\?+1)(\?\?+1)(\?\?+1), Acta Arith. 79 (1997), no. 2, 163 – 171. · Zbl 0869.11071 [5] K. Győry, A. Sárközy, and C. L. Stewart, On the number of prime factors of integers of the form \?\?+1, Acta Arith. 74 (1996), no. 4, 365 – 385. · Zbl 0857.11047 [6] Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. · Zbl 0528.14013 [7] Wolfgang M. Schmidt, Diophantine approximation, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. · Zbl 0421.10019 [8] Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, vol. 1467, Springer-Verlag, Berlin, 1991. · Zbl 0754.11020 [9] C. L. Stewart and R. Tijdeman, On the greatest prime factor of (\?\?+1)(\?\?+1)(\?\?+1), Acta Arith. 79 (1997), no. 1, 93 – 101. · Zbl 0869.11072 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.