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4
functional equation on Z
functional equation on Z
Source:
May 6, 2012
function
induction
algebra proposed
algebra
Problem Statement
Find all functions
f
:
Z
→
Z
f:\mathbb{Z}\rightarrow \mathbb{Z}
f
:
Z
→
Z
satisfying the following:
i
)
i)
i
)
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
;
i
i
)
ii)
ii
)
f
(
m
+
n
)
(
f
(
m
)
−
f
(
n
)
)
=
f
(
m
−
n
)
(
f
(
m
)
+
f
(
n
)
)
f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))
f
(
m
+
n
)
(
f
(
m
)
−
f
(
n
))
=
f
(
m
−
n
)
(
f
(
m
)
+
f
(
n
))
for all
m
,
n
∈
Z
m,n \in \mathbb{Z}
m
,
n
∈
Z
.
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