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Miklós Schweitzer
2007 Miklós Schweitzer
3
Miklós Schweitzer 2007 Problem 3
Miklós Schweitzer 2007 Problem 3
Source:
March 3, 2017
Miklos Schweitzer
number theory
Problem Statement
Denote by
ω
(
n
)
\omega (n)
ω
(
n
)
the number of prime divisors of the natural number
n
n
n
(without multiplicities). Let
F
(
x
)
=
max
n
≤
x
ω
(
n
)
G
(
x
)
=
max
n
≤
x
(
ω
(
n
)
+
ω
(
n
2
+
1
)
)
F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)
F
(
x
)
=
n
≤
x
max
ω
(
n
)
G
(
x
)
=
n
≤
x
max
(
ω
(
n
)
+
ω
(
n
2
+
1
)
)
Prove that
G
(
x
)
−
F
(
x
)
→
∞
G(x)-F(x)\to \infty
G
(
x
)
−
F
(
x
)
→
∞
as
x
→
∞
x\to\infty
x
→
∞
.(translated by Miklós Maróti)
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