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On a Diophantine Equation Involving Lucas Numbers

Seyran S. Ibrahimov, Nazim I. Mahmudov

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.

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.

Paper Structure

This paper contains 4 sections, 7 theorems, 73 equations.

Table of Contents

  1. INTRODUCTION
  2. AUXILIARY RESULTS
  3. MAIN RESULTS
  4. COMMENTS

Key Result

Lemma 2.1

If $r-m$ is even, then:

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Carmichael Primitive Divisor Theorem
  • Lemma 2.4
  • Definition 2.1
  • Lemma 2.5
  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1