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

TL;DR

This paper solves a specific Diophantine equation involving balancing-like sequences defined by a recurrence relation, providing a complete characterization of solutions for fixed parameters and coefficients.

Contribution

It introduces a method to find all solutions of a particular Diophantine equation involving balancing-like sequences associated with the Pillai-Tijdeman problem.

Findings

Complete solution characterization for fixed A and coefficients.

Explicit solutions for the Diophantine equation.

Conditions for non-trivial solutions identified.

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 $C_{1} C_{2} C_{3} \neq 0$.