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Bulgaria National Olympiad
2020 Bulgaria National Olympiad
P3
Integer sequence and Coprimity
Integer sequence and Coprimity
Source: Bulgaria National Olympiad 2020
June 30, 2020
number theory
Integer sequence
Problem Statement
Let
a
1
∈
Z
a_1\in\mathbb{Z}
a
1
∈
Z
,
a
2
=
a
1
2
−
a
1
−
1
a_2=a_1^2-a_1-1
a
2
=
a
1
2
−
a
1
−
1
,
…
\dots
…
,
a
n
+
1
=
a
n
2
−
a
n
−
1
a_{n+1}=a_n^2-a_n-1
a
n
+
1
=
a
n
2
−
a
n
−
1
. Prove that
a
n
+
1
a_{n+1}
a
n
+
1
and
2
n
+
1
2n+1
2
n
+
1
are coprime.
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