Diophantine Equation with Balancing-like Sequences Associated to the Pillai-Tijdeman-type Problem

Bijan Kumar Patel; Prashant Tiwari

paper

Diophantine Equation with Balancing-like Sequences Associated to the Pillai-Tijdeman-type Problem

arXiv:2105.15127

Abstract

Let ${x_{n}}_{n \geq 0}$ be the balancing-like sequence defined by $x_{n + 1} = A x_{n} - x_{n - 1}$ , for $A > 2$ , where $x_{0} = 0$ and $x_{1} = 1$ . In this paper, we demonstrate how to find all the solutions of the Diophantine equation, $C_{1} x_{n_{1}} + C_{2} x_{n_{2}} + C_{3} x_{n_{3}} = C_{4} x_{n_{4}} + C_{5} x_{n_{5}} + C_{6} x_{n_{6}}$ , in fixed integer $A \geq 3$ , $n_{1} > n_{2} > n_{3} \geq 0, n_{4} > n_{5} > n_{6} \geq 0,$ and $C_{1} x_{n_{1}} \neq = C_{4} x_{n_{4}}$ , where $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}$ are given integers such that $C_{1} C_{2} C_{3} \neq = 0$ .

12 pages. arXiv admin note: text overlap with arXiv:2105.01569