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

de Weger, B. M. M.

doi:10.1090/S0025-5718-1986-0856716-7

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

by B. M. M. de Weger PDF

Math. Comp. 47 (1986), 729-739 Request permission

Abstract:

In Part I the diophantine equation ${G_n} = wp_1^{{m_1}} \cdots p_t^{{m_t}}$ was studied, where $\{ {G_n}\} _{n = 0}^\infty$ 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.

References

A. Baker and H. Davenport, The equations $3x^{2}-2=y^{2}$ and $8x^{2}-7=z^{2}$, Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. MR 248079, DOI 10.1093/qmath/20.1.129
Péter Kiss, Zero terms in second-order linear recurrences, Math. Sem. Notes Kobe Univ. 7 (1979), no. 1, 145–152. MR 544926

Mathematika B (Leiden)

A. Pethö and B. M. M. de Weger, Products of prime powers in binary recurrence sequences. I. The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Math. Comp. 47 (1986), no. 176, 713–727. MR 856715, DOI 10.1090/S0025-5718-1986-0856715-5
R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational methods in number theory, Part II, Math. Centre Tracts, vol. 155, Math. Centrum, Amsterdam, 1982, pp. 321–369. MR 702521
Michel Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257–283. MR 598881, DOI 10.4064/aa-37-1-257-283