On binary recurrence sequences. (English) Zbl 0853.11014

Yu, Kunrui; Hung, Ling-kei

Indag. Math., New Ser. 6, No. 3, 341-354 (1995).

Let \(a\), \(b\) be algebraic numbers and \(\lambda\), \(\mu\) be algebraic integers with \(ab \lambda \mu\neq 0\), and let the algebraic number field \(K= \mathbb{Q} (a, b, \lambda, \mu)\) be of degree \(d\) over \(\mathbb{Q}\). Suppose that \(\lambda/ \mu\) is not a root of unity and define \(x_m= a\lambda^m+ b\mu^m\) for \(m\geq 0\), \(m\in \mathbb{N}\). For \(\alpha, \beta\in K\) write \((\alpha)\), \((\alpha, \beta)\) for the fractional ideals generated by \(\alpha\) and by \(\alpha\), \(\beta\), respectively. For a fractional ideal \({\mathcal A}\) in \(K\) and a prime ideal \(\wp\), denote by \(\text{ord}_\wp {\mathcal A}\) the exponent of \(\wp\) in the prime ideal decomposition of \({\mathcal A}\). Let \(P({\mathcal A})\) (resp. \(\mathbb{Q} ({\mathcal A})\)) be the maximum of primes \(p\) lying below some prime ideal \(\wp\) at which \(\text{ord}_\wp {\mathcal A}> 0\) (resp. the product of all distinct rational primes lying below some prime ideals \(\wp\) at which \(\text{ord}_\wp {\mathcal A}>0\)). Finally, let \(h_0\) be the least positive rational integer such that \((\lambda^{h_0}, \mu^{h_0})\) is a principal ideal.
Generalizing as well as refining earlier results of C. L. Stewart [J. Reine Angew. Math. 333, 12-31 (1982; Zbl 0475.10009)] and the reviewer [Indag. Math., New Ser. 1, 85-93 (1990; Zbl 0704.11008)], the authors prove that there exist effectively computable constants \(C_1\) and \(C_2\) depending only on \(d\), \(a\), \(b\), \(\lambda\), \(\mu\) and \(h_0\) such that \[ P\biggl( {{(x_m)} \over {(x_m, x_n)}} \biggr)> C_1 m^{1/( d+1)} \] whenever \(m> n\geq 0\), \(x_n\neq 0\) with \(m> C_2\). Similarly, there exist effectively computable constants \(C_3, \dots, C_6\) depending only on \(d\), \(a\), \(b\), \(\lambda\), \(\mu\) and \(h_0\) such that \[ P((x_m))> C_3 m^{1/ (d+1)}, \qquad Q((x_m))> C_5 \biggl( {m\over {\log m}} \biggr)^{1/d} \] whenever \(m>C_4\) and \(m> C_6\), respectively.

Reviewer: A.Pethö (Debrecen)

MSC:

11B37	Recurrences
11J86	Linear forms in logarithms; Baker’s method

Keywords:

linear recurrent sequences; linear forms in \(p\)-adic logarithms; greatest prime factor; greatest square-free divisor; binary recurrence sequences

Citations:

Zbl 0475.10009; Zbl 0704.11008

References:

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