Multiplicative relations on binary recurrences

Volume 161 / 2013

Florian Luca, Volker Ziegler 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.

Authors

  • Florian LucaMathematical Institute, UNAM
    Mexico, DF, 04510, Mexico
    e-mail
  • Volker ZieglerJohann Radon Institute for
    Computational and Applied Mathematics (RICAM)
    Austrian Academy of Sciences
    Altenbergerstr. 69
    A-4040 Linz, Austria
    e-mail

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