Turkish Journal of Mathematics
DOI
10.3906/mat-2106-86
Abstract
For $k\geq 2$, consider the $k$-Fibonacci sequence $(F_n^{(k)})_{n\geq 2-k}$ having initial conditions $0, \ldots, 0, 1$ ($k$ terms) and each term afterwards is the sum of the preceding $k$ terms. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of $(F_n^{(k)})_{n\geq 2-k}$ with $k=2$ and Tribonacci sequence is $(F_n^{(k)})_{n\geq 2-k}$ with $k=3$. In this paper, we use Baker's method to show that 4, 16, 64, 208, 976, and 1936 are all $k$-Fibonacci numbers of the form $(3^a\pm 1)(3^b\pm 1)$, where $a$ and $b$ are nonnegative integers.
Keywords
$k$-Fibonacci numbers, linear form in logarithms, reduction method
First Page
2664
Last Page
2677
Recommended Citation
GUEYE, ALIOUNE; RIHANE, SALAH EDDINE; and TOGBE, ALAIN
(2021)
"An exponential equation involving $k$-Fibonacci numbers,"
Turkish Journal of Mathematics: Vol. 45:
No.
6, Article 19.
https://doi.org/10.3906/mat-2106-86
Available at:
https://journals.tubitak.gov.tr/math/vol45/iss6/19
