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7
The product is integer [ILL 1974]
The product is integer [ILL 1974]
Source:
January 2, 2011
number theory proposed
number theory
Problem Statement
Let
p
p
p
be a prime number and
n
n
n
a positive integer. Prove that the product
N
=
1
p
n
2
∏
i
=
1
;
2
∤
i
2
n
−
1
[
(
(
p
−
1
)
!
)
(
p
2
i
p
i
)
]
{N=\frac{1}{p^{n^2}}} \prod_{i=1;2 \nmid i}^{2n-1} \biggl[ \left( (p-1)! \right) \binom{p^2 i}{pi}\biggr]
N
=
p
n
2
1
i
=
1
;
2
∤
i
∏
2
n
−
1
[
(
(
p
−
1
)!
)
(
p
i
p
2
i
)
]
Is a positive integer that is not divisible by
p
.
p.
p
.
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