Abstract
AbstractLet $$r\ge 1$$
r
≥
1
be an integer and $$\mathbf{U}:=(U_{n})_{n\ge 0} $$
U
:
=
(
U
n
)
n
≥
0
be the Lucas sequence given by $$U_0=0$$
U
0
=
0
, $$U_1=1, $$
U
1
=
1
,
and $$U_{n+2}=rU_{n+1}+U_n$$
U
n
+
2
=
r
U
n
+
1
+
U
n
, for all $$ n\ge 0 $$
n
≥
0
. In this paper, we show that there are no positive integers $$r\ge 3,~x\ne 2,~n\ge 1$$
r
≥
3
,
x
≠
2
,
n
≥
1
such that $$U_n^x+U_{n+1}^x$$
U
n
x
+
U
n
+
1
x
is a member of $$\mathbf{U}$$
U
.
Funder
Austrian Science Fund
University of the Witwatersrand, Johannesburg
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
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