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Iran Team Selection Test
2009 Iran Team Selection Test
11
Iran TST 2009-Day4-P2
Iran TST 2009-Day4-P2
Source:
May 17, 2009
modular arithmetic
number theory proposed
number theory
Problem Statement
Let
n
n
n
be a positive integer. Prove that
3
5
2
n
−
1
2
n
+
2
≡
(
−
5
)
3
2
n
−
1
2
n
+
2
(
m
o
d
2
n
+
4
)
.
3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}.
3
2
n
+
2
5
2
n
−
1
≡
(
−
5
)
2
n
+
2
3
2
n
−
1
(
mod
2
n
+
4
)
.
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