The Ramanujan Journal

, Volume 46, Issue 1, pp 49–75

# Diophantine equations with products of consecutive members of binary recurrences

• Attila Bérczes
• Yuri F. Bilu
• Florian Luca
Article

## Abstract

We prove a finiteness result for the number of solutions of a Diophantine equation of the form $$u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2$$, where $$\{ u_n\}_{n\ge 1}$$ is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.

## Keywords

Diophantine equations Binary recurrences Applications of linear forms in logarithms

11B39 11D61

## Notes

### Acknowledgements

We thank the referee for careful reading and detecting a flaw in the initial version. We also thank Karim Belabas and Michael Mossinghoff for helpful suggestions.

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## Authors and Affiliations

• Attila Bérczes
• 1
• Yuri F. Bilu
• 2
• Florian Luca
• 3
• 4
• 5
1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
2. 2.Institut de Mathématiques de BordeauxUniversité de Bordeaux and CNRSTalenceFrance
3. 3.School of MathematicsUniversity of the WitwatersrandWitsSouth Africa
4. 4.Max Planck Institute for MathematicsBonnGermany
5. 5.Department of Mathematics, Faculty of SciencesUniversity of OstravaOstrava 1Czech Republic