Pillai-Type Equations with Lucas Numbers and S-Unit Solutions

Abstract

In this paper, we investigate the exponential Diophantine equation $
L_n - 5^x 7^y = c,$ where \(L_n\) denotes the \(n\)-th Lucas number. The Lucas sequence is defined by the initial values
\(L_0 = 2\), \(L_1 = 1\), and the recurrence relation
\(L_{n+2} = L_{n+1} + L_n\) for all \(n \geq 0\).
We show that when \(c = 0\), the equation admits exactly two distinct solutions.
Moreover, for any \(c \in \mathbb{N}\), we prove that there is no integer \(c\) for which the equation
has at least three distinct solutions \((n, x, y) \in \mathbb{Z}_{\geq 0}^3\).