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arXiv · 2105.15127

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

Bijan Kumar Patel·Prashant Tiwari
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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 $C_{1} C_{2} C_{3} \neq 0$.

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Bijan Kumar Patel, Prashant Tiwari. 2021-05-31. Diophantine Equation with Balancing-like Sequences Associated to the Pillai-Tijdeman-type Problem. https://arxiv.org/abs/2105.15127

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