Multiplicative relations on binary recurrences

Florian Luca, and Volker Ziegler. "Multiplicative relations on binary recurrences." Acta Arithmetica 161.2 (2013): 183-199. <http://eudml.org/doc/279204>.

@article{FlorianLuca2013,
abstract = {Given a binary recurrence $\{u_n\}_\{n≥0\}$, we consider the Diophantine equation $u_\{n_1\}^\{x_1\} ⋯ u_\{n_L\}^\{x_L\} = 1$ with nonnegative integer unknowns $n_1,..., n_L$, where $n_i ≠ n_j$ for 1 ≤ i < j ≤ L, $max\{|x_i|: 1 ≤ i ≤ L\} ≤ K$, 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.},
author = {Florian Luca, Volker Ziegler},
journal = {Acta Arithmetica},
keywords = {binary recurrences; Diophantine equations; Diophantine inequalities},
language = {eng},
number = {2},
pages = {183-199},
title = {Multiplicative relations on binary recurrences},
url = {http://eudml.org/doc/279204},
volume = {161},
year = {2013},
}

TY - JOUR
AU - Florian Luca
AU - Volker Ziegler
TI - Multiplicative relations on binary recurrences
JO - Acta Arithmetica
PY - 2013
VL - 161
IS - 2
SP - 183
EP - 199
AB - Given a binary recurrence ${u_n}_{n≥0}$, we consider the Diophantine equation $u_{n_1}^{x_1} ⋯ u_{n_L}^{x_L} = 1$ with nonnegative integer unknowns $n_1,..., n_L$, where $n_i ≠ n_j$ for 1 ≤ i < j ≤ L, $max{|x_i|: 1 ≤ i ≤ L} ≤ K$, 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.
LA - eng
KW - binary recurrences; Diophantine equations; Diophantine inequalities
UR - http://eudml.org/doc/279204
ER -