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China Girls Math Olympiad
2019 China Girls Math Olympiad
5
5
Part of
2019 China Girls Math Olympiad
Problems
(1)
Chinese Girls Mathematical Olympiad 2019, Problem 5
Source: Chinese Girls Mathematical Olympiad 2019 Day 2 P1
8/13/2019
Let
p
p
p
be a prime number such that
p
∣
(
2
2019
−
1
)
.
p\mid (2^{2019}-1) .
p
∣
(
2
2019
−
1
)
.
The sequence
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
satisfies the following conditions:
a
0
=
2
,
a
1
=
1
,
a
n
+
1
=
a
n
+
p
2
−
1
4
a
n
−
1
a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}
a
0
=
2
,
a
1
=
1
,
a
n
+
1
=
a
n
+
4
p
2
−
1
a
n
−
1
(
n
≥
1
)
.
(n\geq 1).
(
n
≥
1
)
.
Prove that
p
∤
(
a
n
+
1
)
,
p\nmid (a_n+1),
p
∤
(
a
n
+
1
)
,
for any
n
≥
0.
n\geq 0.
n
≥
0.
number theory
algebra
China