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Problems
Contests
National and Regional Contests
China Contests
China Western Mathematical Olympiad
2015 China Western Mathematical Olympiad
8
8
Part of
2015 China Western Mathematical Olympiad
Problems
(1)
Fermat prime
Source: CWMI 2015 Q8
8/20/2015
Let
k
k
k
be a positive integer, and
n
=
(
2
k
)
!
n=\left(2^k\right)!
n
=
(
2
k
)
!
.Prove that
σ
(
n
)
\sigma(n)
σ
(
n
)
has at least a prime divisor larger than
2
k
2^k
2
k
, where
σ
(
n
)
\sigma(n)
σ
(
n
)
is the sum of all positive divisors of
n
n
n
.
number theory