Multiplicative relations on binary recurrences

Florian Luca ¹   ; Volker Ziegler ²

¹ Mathematical Institute, UNAM Mexico, DF, 04510, Mexico
² Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences Altenbergerstr. 69 A-4040 Linz, Austria

Acta Arithmetica, Tome 161 (2013) no. 2, pp. 183-199

Résumé

Given a binary recurrence $\{u_n\}_{n\ge 0}$, we consider the Diophantine equation $$ u_{n_1}^{x_1} \cdots u_{n_L}^{x_L}=1 $$ with nonnegative integer unknowns $n_1,\ldots ,n_L$, where $n_i\not =n_j$ for $1\le i j\le L$, $\max\{|x_i|: 1\le i\le L\}\leq K$, and $K$ is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

Détail

DOI : 10.4064/aa161-2-4

Keywords: given binary recurrence consider diophantine equation cdots nonnegative integer unknowns ldots where max leq fixed parameter above equation has only finitely many solutions largest explicitly bounded demonstrate strength method completely solving particular diophantine equation above form