MathDB
Problems
Contests
National and Regional Contests
Malaysia Contests
Malaysian IMO Training Camp
2024 Malaysian IMO Training Camp
7
7
Part of
2024 Malaysian IMO Training Camp
Problems
(1)
max v_p are equal in consecutive values
Source: Malaysian SST 2024 P7
9/5/2024
Let
P
P
P
be the set of all primes. Given any positive integer
n
n
n
, define
f
(
n
)
=
max
p
∈
P
v
p
(
n
)
\displaystyle f(n) = \max_{p \in P}v_p(n)
f
(
n
)
=
p
∈
P
max
v
p
(
n
)
Prove that for any positive integer
k
≥
2
k\ge 2
k
≥
2
, there exists infinitely many positive integers
m
m
m
such that
f
(
m
+
1
)
=
f
(
m
+
2
)
=
⋯
=
f
(
m
+
k
)
f(m+1) = f(m+2) = \cdots = f(m+k)
f
(
m
+
1
)
=
f
(
m
+
2
)
=
⋯
=
f
(
m
+
k
)
Proposed by Ivan Chan Guan Yu
number theory