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National and Regional Contests
PEN Problems
PEN D Problems
3
D 3
D 3
Source:
May 25, 2007
modular arithmetic
number theory
prime numbers
Congruences
Problem Statement
Show that
(
−
1
)
p
−
1
2
(
p
−
1
p
−
1
2
)
≡
4
p
−
1
(
m
o
d
p
3
)
(-1)^{\frac{p-1}{2}}{p-1 \choose{\frac{p-1}{2}}}\equiv 4^{p-1}\pmod{p^{3}}
(
−
1
)
2
p
−
1
(
2
p
−
1
p
−
1
)
≡
4
p
−
1
(
mod
p
3
)
for all prime numbers
p
p
p
with
p
≥
5
p \ge 5
p
≥
5
.
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