Nonnegative integer solutions of the equation $L_n^{(k)} - L_m^{(k)} = 2\cdot 3^a $

Bakr Kourouma; Salah Eddine Rihane; Alain Togbe

Vol. 30 No. 1 (2025)

Regular Articles

Nonnegative integer solutions of the equation $L_n^{(k)} - L_m^{(k)} = 2\cdot 3^a $

Published 2025-03-11

Bakr Kourouma
Salah Eddine Rihane
Alain Togbe

Bakr Kourouma
Mathematics Department, Faculty of Science, Gamal Abdel Nasser University, Conakry

Salah Eddine Rihane
National Higher School of Mathematics, Scientific and Technology Hub of Sidi Abdellah

Alain Togbe
Purdue University Northwest

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Abstract

For an integer $k \geq 2,$ let $(L_n^{(k)} )_n$ be the $k$-generalized Lucas sequence, which starts with $0,...,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In $2019,$ B. D. Bitim found all the solutions of the Diophantine equation $L_n-L_m=2\times 3^a$. In this paper, we generalize this result by considering the $k$-generalized Lucas sequence, i.e., we study the Diophantine equation $L_n^{(k)} - L_m^{(k)} = 2\times 3^a $ in positive integers $n, m, a$ with $k \geq 3$. To obtain our main result, we use Baker's method and Baker-Davenport reduction method.

Keywords

Generalized Lucas numbers, linear forms in complex logarithms, and reduction method

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Supplementary File(s)

5307-15440-1-SP - final

Author Biography

Alain Togbe

Professor