Multiplicative relations on binary recurrences

Florian Luca; Volker Ziegler

Acta Arithmetica, Tome 161 (2013), p. 183-199 / Harvested from The Polish Digital Mathematics Library

Résumé

Given a binary recurrence ${u_{n}}_{n \geq 0}$ , we consider the Diophantine equation $u_{n_{1}}^{x_{1}} \dots u_{n_{L}}^{x_{L}} = 1$ with nonnegative integer unknowns $n_{1}, . . ., n_{L}$ , where $n_{i} \neq n_{j}$ for 1 ≤ i < j ≤ L, $m a x | x_{i} | : 1 \leq i \leq L \leq K$ , 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.

Publié le : 2013-01-01

Zbl 1316.11012

EUDML-ID : urn:eudml:doc:279204

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     author = {Florian Luca and Volker Ziegler},
     title = {Multiplicative relations on binary recurrences},
     journal = {Acta Arithmetica},
     volume = {161},
     year = {2013},
     pages = {183-199},
     zbl = {1316.11012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-2-4}
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Florian Luca; Volker Ziegler. Multiplicative relations on binary recurrences. Acta Arithmetica, Tome 161 (2013) pp. 183-199. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-2-4/