Products of prime powers in binary recurrence sequences. II. The elliptic case, with an application to a mixed quadratic-exponential equation

Author:
B. M. M. de Weger

Journal:
Math. Comp. **47** (1986), 729-739

MSC:
Primary 11D61; Secondary 11Y50

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856716-7

MathSciNet review:
856716

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Abstract: In Part I the diophantine equation was studied, where is a linear binary recurrence sequence with positive discriminant. In this second part we extend this to negative discriminants. We use the *p*-adic and complex Gelfond-Baker theory to find explicit upper bounds for the solutions of the equation. We give algorithms to reduce those bounds, based on diophantine approximation techniques. Thus we have a method to solve the equation completely for arbitrary values of the parameters. We give an application to a quadratic-exponential equation.

**[1]**A. Baker & H. Davenport, "The equations and ,"*Quart. J. Math. Oxford Ser.*(2), v. 20, 1969, pp. 129-137. MR**0248079 (40:1333)****[2]**P. Kiss, "Zero terms in second order linear recurrences,"*Math. Sem. Notes Kobe Univ.*, v. 7, 1979, pp. 145-152. MR**544926 (80j:10015)****[3]**K. Mahler, "Eine arithmetische Eigenschaft der rekurrierenden Reihen,"*Mathematika B (Leiden)*, v. 3, 1934, pp. 153-156.**[4]**A. Pethö & B. M. M. de Weger, "Products of prime powers in binary recurrence sequences. I,"*Math. Comp.*, v. 47, 1986, pp. 713-727. MR**856715 (87m:11027a)****[5]**R. J. Stroeker & R. Tijdeman, "Diophantine equations," in*Computational Methods in Number Theory*(H. W. Lenstra, Jr. and R. Tijdeman, eds.), MC Tract 155, Amsterdam, 1982, pp. 321-369. MR**702521 (84i:10014)****[6]**M. Waldschmidt, "A lower bound for linear forms in logarithms,"*Acta Arith.*, v. 37, 1980, pp. 257-283. MR**598881 (82h:10049)**

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0856716-7

Article copyright:
© Copyright 1986
American Mathematical Society