.svg){kind=logo}
NARAYANA’S COWS AND TRIBONACCI SEQUENCES IN PILLAI’S PROBLEM
Keywords:
Diophantine equations, Tribonacci sequence, Narayana’s cows sequence, linear forms in logarithms, reduction method, continued fractionsDOI:
https://doi.org/10.17654/0972555525017Abstract
Let $\left(N_n\right)_{n\ge0}$ be the Narayana's cows sequence and $\left(T_n\right)_{n\ge0}$ the Tribonacci sequence. Then we study and completely solve the Diophantine equation $N_m - T_n = N_{m_1}-T_{n_1}$ where $m, n, m_1$ and $n_1$ are
positive integers with $(m, n)\neq (m_1, n_1)$.
Received: December 23, 2024
Revised: March 5, 2025
Accepted: March 21, 2025
References
A. Baker and H. Davenport, The equations and Quart. J. Math. Oxford Ser. 20 (1969), 129-137.
S. G. Sanchez and F. Luca, Linear combinations of factorials and S-units in a binary recurrence sequence, Ann. Math. Quebec 38 (2014), 169-188.
M. Ddamulira, F. Luca and M. Rakotomalala, On a problem of Pillai with Fibonacci and powers of 2, Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017), 411-421.
A. C. G. Lomelia, S. H. Hernandeza and F. Luca, Pillai’s problem with the Fibonacci and Padovan sequences, Ann. Math. Inform. 50 (2019), 101-115.
Pagdame Tiebekabe and Serge Adonsou, On Pillai’s problem involving two linear recurrent sequences: Padovan and Fibonacci, Malaya J. Mat. 10(3) (2022), 204-215. http://doi.org/10.26637/mjm1003/003.
Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math. 163(2) (2006), 969-1018.
J. J. Bravo, F. Luca and K. Yazan, On Pillai’s problem with Tribonacci numbers and powers of 2, Bull. Korean Math. Soc 54(3) (2017), 1069-1080.
A. C. G. Lomeli and S. H. Hernandez, Pillai’s problem with Tribonacci numbers and powers of two, Rev. Colombiana Mat. 53(1) (2019), 1-14.
K. Chim, I. Pink and V. Ziegler, On a variant of Pillai’s problem, Int. J. Number Theory 7 (2017), 1711-1727.
K. C. Chim, I. Pink and V. Ziegler, On a variant of Pillai’s problem II, J. Number Theory 183 (2018), 269-290.
E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), 125-180.
A. Dujella and A. Petho, A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford 49(3) (1998), 291-306.
S. H. Hernandez, F. Luca and L. M. Rivera, On Pillai’s problem with the Fibonacci and Pell sequences, Bol. Soc. Mat. Mex. (3) 25 (2019), 495-507.
S. S. Pillai, On J. Indian Math. Soc. 2 (1936), 119-122.
S. S. Pillai, On the equation Bull. Calcutta Math. Soc. 37 (1945), 15-20.
R. J. Stroeker and R. Tijdeman, Diophantine equations, Computational Methods in Number Theory, Part II, Math. Centre Tracts, Math. Centrum, Amsterdam, 1982, pp. 321-369.
T. Pagdame, A. Serge and D. Ismaila, On solutions of the Diophantine equation Moroccan J. Algebra Geom. Appl. 2(2) (2023), 246-261.
S. Heintze and V. Ziegler, On Pillai’s problem involving Lucas sequences of the second kind, Res. Number Theory 10 (2024), Article number 51.
https://doi.org/10.1007/s40993-024-00534-5.
M. Ddamulira, On a problem of Pillai with Fibonacci numbers and powers of 3, Bol. Soc. Mat. Mex. 26(2) (2020), 263-277. doi: 10.1007/s40590-019-00263-1.
Garcia Lomeli, Ana Cecilia and Santos Hernandez Hernandez, Pillai’s problem with Padovan numbers and powers of two, Rev. Colombiana Mat. 53(1) (2019), 1-14.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
