Multiplicative relations on binary recurrences
Volume 161 / 2013
Acta Arithmetica 161 (2013), 183-199
MSC: 11B37, 11D57, 11D75, 11J25.
DOI: 10.4064/aa161-2-4
Abstract
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.
