On the Diophantine equation $$L_{n}-L_{m}=2\cdot 3^{a}$$

Abstract

In this paper we find (n, m, a) solutions of the Diophantine equation \(L_{n}-L_{m}=2\cdot 3^{a}\), where \(L_{n}\) and \(L_{m}\) are Lucas numbers with \(a\ge 0\) and \(n>m\ge 0\). For proving our theorem, we use lower bounds for linear forms in logarithms and Baker–Davenport reduction method in Diophantine approximation.