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On the largest prime factor of \((ab+1)(ac+1)(bc+1)\). (English) Zbl 1108.11030

Fix a finite set of primes \({\mathcal P}\), and let \({\mathcal S}\) be the set of all non-zero integers whose prime factors all belong to \({\mathcal P}\). The authors prove that for every \(\varepsilon > 0\) there exists a constant \(c_{\varepsilon}\) such that whenever \(s_1, s_2 \in {\mathcal S}\) with \(| s_1| \) and \(| s_2| \) multiplicatively independent, with \(\min\{| s_1| ,| s_2| \} > 1\), and with \(\max\{| s_1| ,| s_2| \} > c_{\varepsilon}\), one has \(\gcd(s_1-1,s_2-1) < (\max\{| s_1| ,| s_2| \})^{\varepsilon}\). This generalizes a result of Y. Bugeaud, P. Corvaja, and U. Zannier [Math. Z. 243, No. 1, 79–84 (2003; Zbl 1021.11001)] on the greatest common divisor of \(a^n-1\) and \(b^n-1\). The proof uses Schmidt’s subspace theorem. Now fix integers \(n > k \geq 2\), suppose that \(a_1 > a_2 > \dots > a_n > 0\), and let \({\mathcal I} = \{i_1, \dots, i_k\}\) denote a subset of \(\{1,\dots, n\}\). The authors use their first theorem to deduce that the largest prime factor of \(\prod_{| {\mathcal I}| =k} (a_{i_1} \cdots a_{i_k} + 1)\) tends to infinity with \(a_1\). In particular, this proves a conjecture of Györy, Sárkozy, and Stewart concerning the largest prime factor of \((ab+1)(ac+1)(bc+1)\). It is noted that the truth of this latter conjecture (the \(n=3\), \(k=2\) case of the authors’ result) was established independently by P. Corvaja and U. Zannier [Proc. Am. Math. Soc. 131, No. 6, 1705–1709 (2003; Zbl 1077.11052)] at about the same time.

MSC:

11D75 Diophantine inequalities
11J61 Approximation in non-Archimedean valuations
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