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Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

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, m a x | 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.

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