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KoMaL A Problems
KoMaL A Problems 2017/2018
A. 706
From Komal A.706
From Komal A.706
Source:
November 16, 2017
komal
function equation
function
algebra
Problem Statement
Find all positive integer
k
k
k
s for which such
f
f
f
exists and unique:
f
(
m
n
)
=
f
(
n
)
f
(
m
)
f(mn)=f(n)f(m)
f
(
mn
)
=
f
(
n
)
f
(
m
)
for
n
,
m
∈
Z
+
n, m \in \mathbb{Z^+}
n
,
m
∈
Z
+
f
n
k
(
n
)
=
n
f^{n^k}(n)=n
f
n
k
(
n
)
=
n
for all
n
∈
Z
+
n \in \mathbb{Z^+}
n
∈
Z
+
for which
f
x
(
n
)
f^x (n)
f
x
(
n
)
means the n times operation of function
f
f
f
(i.e.
f
(
f
(
.
.
.
f
(
n
)
)
.
.
.
)
f(f(...f(n))...)
f
(
f
(
...
f
(
n
))
...
)
)
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