On a Diophantine Equation Involving Lucas Numbers

Abstract

Let L_t denote the t-th Lucas number. We prove that the Diophantine equation L_m^{n+k} + L_m^n = L_r has no solutions in positive integers r, m, n, and k with m >= 2. In the case n = 1, the proof is based on a precise factorization formula for the difference of two Lucas numbers and the Carmichael Primitive Divisor Theorem. For n >= 2, we apply lower bounds for linear forms in logarithms due to Matveev, combined with Legendre's lemma, an exact divisibility property for powers of Lucas numbers, and computer-assisted computations to complete the proof.