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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## Multiplicative relations on binary recurrences

### Tom 161 / 2013

Acta Arithmetica 161 (2013), 183-199 MSC: 11B37, 11D57, 11D75, 11J25. DOI: 10.4064/aa161-2-4

#### Streszczenie

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.

#### Autorzy

• Florian LucaMathematical Institute, UNAM
Mexico, DF, 04510, Mexico
e-mail
• Volker ZieglerJohann Radon Institute for
Computational and Applied Mathematics (RICAM)