On solutions of the Diophantine equation \mathcal{P}_m − L_n = c
Pagdame Tiebekabe\,^{1,2} , Serge Adonsou\,^3 and Ismaïla Diouf \,^1
\,^1 Département de Mathématiques et Informatique, Université Cheikh Anta Diop, Dakar, BP 5005 Dakar-fann, Sénégal.
\,^2 Département de Mathématiques, Université de Kara, BP 404 Kara – Togo.
\,^3 Département de Mathématique et Informatique, African Insitute for Mathematical Sciences, 6 Melrose Avenue Cape Town 7945.
Pages 246–261 | Received 03 March 2023, Accepted 12 August 2023, Published 11 December 2023
Abstract
In this article, we determine all the integers c having at least two representations as difference between two linear recurrent sequences. This is a variant of the Pillai’s equation. This equation is an exponential Diophantine equation. The proof of our main theorem uses lower bounds for linear forms of logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.
Keywords: Linear forms in logarithms, Diophantine equations, Pillai’s problem, Linear recurrent sequences.
MSC numbers: Primary 11B39, 11J86; Secondary 11D617.
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