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National and Regional Contests
China Contests
China Team Selection Test
2003 China Team Selection Test
3
A sequence
A sequence
Source: China TST 2003
June 29, 2006
number theory unsolved
number theory
Problem Statement
Sequence
{
a
n
}
\{ a_n \}
{
a
n
}
satisfies:
a
1
=
3
a_1=3
a
1
=
3
,
a
2
=
7
a_2=7
a
2
=
7
,
a
n
2
+
5
=
a
n
−
1
a
n
+
1
a_n^2+5=a_{n-1}a_{n+1}
a
n
2
+
5
=
a
n
−
1
a
n
+
1
,
n
≥
2
n \geq 2
n
≥
2
. If
a
n
+
(
−
1
)
n
a_n+(-1)^n
a
n
+
(
−
1
)
n
is prime, prove that there exists a nonnegative integer
m
m
m
such that
n
=
3
m
n=3^m
n
=
3
m
.
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