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1971 IMO Longlists
53
53
Part of
1971 IMO Longlists
Problems
(1)
The limit of the multiplicity of p [ILL 1971]
Source:
1/1/2011
Denote by
x
n
(
p
)
x_n(p)
x
n
(
p
)
the multiplicity of the prime
p
p
p
in the canonical representation of the number
n
!
n!
n
!
as a product of primes. Prove that
x
n
(
p
)
n
<
1
p
−
1
\frac{x_n(p)}{n}<\frac{1}{p-1}
n
x
n
(
p
)
<
p
−
1
1
and
lim
n
→
∞
x
n
(
p
)
n
=
1
p
−
1
\lim_{n \to \infty}\frac{x_n(p)}{n}=\frac{1}{p-1}
lim
n
→
∞
n
x
n
(
p
)
=
p
−
1
1
.
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