Abstract
AbstractThe well-known Fibonacci sequence has several generalizations, among them, the k-generalized Fibonacci sequence denoted by $$F^{(k)}$$
F
(
k
)
. The first k terms of this generalization are $$0, \ldots , 0, 1$$
0
,
…
,
0
,
1
and each one afterward corresponds to the sum of the preceding k terms. For the Fibonacci sequence the formula $$F_{n+1}^2 - F_{n-1}^2 = F_{2n}$$
F
n
+
1
2
-
F
n
-
1
2
=
F
2
n
holds for every $$n \ge 1$$
n
≥
1
. In this paper, we study the above identity on the k-generalized Fibonacci sequence terms, completing the work done by Bensella et al. (On the exponential Diophantine equation $$(F_{m+1}^{(k)})^x - (F_{m-1}^{(k)})^x = F_n^{(k)}$$
(
F
m
+
1
(
k
)
)
x
-
(
F
m
-
1
(
k
)
)
x
=
F
n
(
k
)
, 2022. arxiv:2205.13168).
Publisher
Springer Science and Business Media LLC
Reference19 articles.
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