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On a conjecture concerning the multiplicity of the Tribonacci sequence

Volume 159 / 2020

Eric F. Bravo, Carlos A. Gómez, Bir Kafle, Florian Luca, Alain Togbé Colloquium Mathematicum 159 (2020), 61-69 MSC: Primary 11B39; Secondary 11J86. DOI: 10.4064/cm7729-2-2019 Published online: 10 October 2019

Abstract

The Tribonacci sequence $\left \{T_{k}\right \}_{k\in {\mathbb Z}}$ is defined by $T_0=0,T_1=T_2=1$ and the recurrence $T_{k+3}=T_{k+2}+T_{k+1}+T_k$ for all $k\in {\mathbb Z}$. In 2016, Kuhapatanakul et al. made a conjecture concerning the positive integer solutions $(m,n)$ of the Diophantine equation $T_{m}=T_{-n}$. We confirm their conjecture.

Authors

  • Eric F. BravoDepartamento de Matemáticas
    Universidad del Valle
    Cll 13 # 100-00
    Cali, Colombia
    e-mail
  • Carlos A. GómezDepartamento de Matemáticas
    Universidad del Valle
    Cll 13 # 100-00
    Cali, Colombia
    e-mail
  • Bir KafleDepartment of Mathematics,
    Statistics and Computer Science
    Purdue University Northwest
    1401 S. U.S. 421
    Westville, IN 46391, U.S.A.
    e-mail
  • Florian LucaSchool of Mathematics
    University of the Witwatersrand
    Private Bag X3
    Wits 2050
    Johannesburg, South Africa
    and
    Department of Mathematics
    Faculty of Science
    University of Ostrava
    30. dubna 22
    701 03 Ostrava 1, Czech Republic
    and
    Research Group in Algebraic Structures
    and Applications
    King Abdulaziz University
    Jeddah, Saudi Arabia
    e-mail
  • Alain TogbéDepartment of Mathematics,
    Statistics and Computer Science
    Purdue University Northwest
    1401 S. U.S. 421
    Westville, IN 46391, U.S.A.
    e-mail

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