## On binary recurrence sequences.(English)Zbl 0853.11014

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.

### MSC:

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

### Citations:

Zbl 0475.10009; Zbl 0704.11008
Full Text:

### References:

 [1] Baker, A.; Wüstholz, G., Logarithmic forms and group varieties, J. reine angew. Math., 442, 19-62 (1993) · Zbl 0788.11026 [2] Mahler, K., Eine arithmetische Eigenschaft der rekurrierenden Reihen, Mathematica B (Leiden), 3, 153-156 (1934) · JFM 60.0919.05 [3] Pethö, A., On the greatest prime factor and divisibility properties of linear recursive sequences, Indag. Mathem. N.S. (1), 1, 85-93 (1990) · Zbl 0704.11008 [4] Pólya, G., Arithmetische Eigenschaften der Reihenentwicklungen rationaler Funktionen, J. reine angew. Math., 151, 1-31 (1921) · JFM 47.0276.02 [5] Rosser, J. B.; Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois J. Math., 6, 64-94 (1962) · Zbl 0122.05001 [6] Shorey, T. N., Linear forms in members of a binary recursive sequence, Acta Arith., 43, 317-331 (1984) · Zbl 0491.10011 [7] Schinzel, A., On two theorems of Gelfond and some of their applications, Acta Arith., 13, 177-236 (1967) · Zbl 0159.07101 [8] Stewart, C. L., Divisor Properties of Arithmetical Sequences, (Ph.D. Thesis (1976), University of Cambridge) [9] Stewart, C. L., On divisors of terms of linear recurrence sequences, J. reine angew. Math., 333, 12-31 (1982) · Zbl 0475.10009 [10] Shorey, T. N.; Tijdeman, R., Exponential Diophantine Equations, (Cambridge Tracts in Mathematics, 87 (1986), Cambridge Univ. Press,: Cambridge Univ. Press, Cambridge) · Zbl 1156.11015 [11] Kunrui, Yu, Linear forms in $$p$$-adic logarithms, Acta Arith., 53, 107-186 (1989) · Zbl 0699.10050 [12] Kunrui, Yu, Linear forms in $$p$$-adic logarithms II, Compositio Math., 74, 15-113 (1990) · Zbl 0723.11034 [13] Kunrui, Yu, Linear forms in $$p$$-adic logarithms III, Compositio Math., 91, 241-276 (1994) · Zbl 0819.11025
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