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Baltic Way
2019 Baltic Way
2
Fibonacci Diophantine
Fibonacci Diophantine
Source: 2019 Baltic Way P2
November 18, 2019
algebra
Problem Statement
Let
(
F
n
)
(F_n)
(
F
n
)
be the sequence defined recursively by
F
1
=
F
2
=
1
F_1=F_2=1
F
1
=
F
2
=
1
and
F
n
+
1
=
F
n
+
F
n
−
1
F_{n+1}=F_n+F_{n-1}
F
n
+
1
=
F
n
+
F
n
−
1
for
n
≥
2
n\geq 2
n
≥
2
. Find all pairs of positive integers
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
5
F
x
−
3
F
y
=
1.
5F_x-3F_y=1.
5
F
x
−
3
F
y
=
1.
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