On a Diophantine Equation Involving Lucas Numbers
Number Theory
2026-02-19 v3
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.
Keywords
Cite
@article{arxiv.2506.14379,
title = {On a Diophantine Equation Involving Lucas Numbers},
author = {Seyran S. Ibrahimov and Nazim I. Mahmudov},
journal= {arXiv preprint arXiv:2506.14379},
year = {2026}
}