Abstract
AbstractLet $$d \ge 2$$
d
≥
2
be an integer which is not a square. We show that if $$(F_n)_{n\ge 0}$$
(
F
n
)
n
≥
0
is the Fibonacci sequence and $$(X_m, Y_m)_{m\ge 1}$$
(
X
m
,
Y
m
)
m
≥
1
is the mth solution of the Pell equation $$X^2 -dY^2 = \pm 1$$
X
2
-
d
Y
2
=
±
1
, then the equation $$Y_m = F_n$$
Y
m
=
F
n
has at most two positive integer solutions (m, n) except for $$d=2$$
d
=
2
when it has three solutions $$(m,n)=(1,2),(2,3),(3,5)$$
(
m
,
n
)
=
(
1
,
2
)
,
(
2
,
3
)
,
(
3
,
5
)
.
Funder
National Research Foundation and The World Academy of Science
Publisher
Springer Science and Business Media LLC
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