Let \((L_n)_{n \ge 0}\) be the Lucas sequence defined by \(L_{n + 2} = L_{n + 1} + L_n\) for all \(n \ge 0\) , with initial conditions \(L_0 = 2\) and \(L_1 = 1\) . In this paper, we find all solutions of the Diophantine equations \(L_{n} - L_{m} = 5 \cdot 2^a\) and \(L_{n} - L_{m} = 2 \cdot 5^a\) , where the parameters n, m and a are nonnegative integers such that \(n > m\) . As a corollary, we also determine all solutions of the equation \(L_{n} - L_{m} = 10^a\) in nonnegative integers (n, m, a) with \(n > m\) . In order to proof our theorems, we use Baker’s method and various properties of Lucas numbers, along with the theory of lower bounds for linear forms in logarithms of algebraic numbers due to Matveev and Dujella-Pethő version of the reduction method of Baker-Davenport in Diophantine approximation.
