Original scientific paper
https://doi.org/10.64785/mc.30.1.7
Nonnegative integer solutions of the equation \(L_{n}^{(k)}- L_{m}^{(k)}=2\times 3^{\alpha }\)
Bakr Kouroumane Rihane, Alain Togbe
orcid.org/0009-0004-5196-6291
; Mathematics Department, Faculty of Science, Gamal Abdel Nasser University, Conakry, Guinea
Salah Eddine Rihane
; National Higher School of Mathematics, Scientific and Technology Hub of Sidi Abdellah, Algeria
Alain Togbe
orcid.org/0000-0002-5882-936X
; Department of Mathematics and Statistics, Purdue University Northwest, 2200 169th Street, Hammond, USA
Abstract
For an integer k ≥ 2, let \( \left (L_{n}^{(k)} \right )_{n}\geq -(k-2)\) 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, Bitim found all the solutions of the Diophantine equation \(L_{n}-L_{m}=2\cdot 3^{\alpha }\). In this paper, we generalize this result by considering the k-generalized
Lucas sequence, i.e., we solve the Diophantine equation \(L_{n}^{(k)}- L_{m}^{(k)}=2\times 3^{\alpha }\) in positive
integers n, m, a with k ≥ 3. To obtain our main result, we use Baker’s method and the Baker-Davenport reduction method.
Keywords
Generalized Lucas numbers; linear forms in complex logarithms; and reduction method
Hrčak ID:
329426
URI
Publication date:
11.3.2025.
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