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.
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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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A.B. was supported by the University of Debrecen, by a János Bolyai Scholarship of the Hungarian Academy of Sciences, and by Grants K100339 and NK104208 of the Hungarian National Foundation for Scientific Research. F. L. worked on this paper when he visited the University of Debrecen in July 2016 and while he visited the Max Planck Institute for Mathematics in Bonn in 2017. He thanks both these institutions for hospitality and support. He was also supported by Grants CPRR160325161141 and an A-rated researcher award both from the NRF of South Africa and by Grant No. 17-02804S of the Czech Granting Agency.
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Bérczes, A., Bilu, Y.F. & Luca, F. Diophantine equations with products of consecutive members of binary recurrences. Ramanujan J 46, 49–75 (2018). https://doi.org/10.1007/s11139-017-9935-0
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DOI: https://doi.org/10.1007/s11139-017-9935-0