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On the greatest prime factor of \((ab+1)(ac+1)\). (English) Zbl 1077.11052

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)].

MSC:

11J25 Diophantine inequalities
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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
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[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
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