On a Diophantine Equation Involving Lucas Numbers

TL;DR

This paper proves that a specific exponential Diophantine equation involving Lucas numbers has no solutions in positive integers, using advanced number theory techniques and computational methods.

Contribution

It establishes the non-existence of solutions for a particular Lucas number equation, combining factorization, primitive divisor theorems, and linear forms in logarithms.

Findings

No solutions in positive integers for the equation involving Lucas numbers.

The proof employs factorization formulas and primitive divisor theorems.

Computational methods support the theoretical proof.

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.