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2015 HMIC
5
2015 HMIC #5: (cyclotomic) Diophantine
2015 HMIC #5: (cyclotomic) Diophantine
Source:
April 26, 2015
HMIC
cyclotomic field
roots of unity
diophantine
Problem Statement
Let
ω
=
e
2
π
i
/
5
\omega = e^{2\pi i /5}
ω
=
e
2
πi
/5
be a primitive fifth root of unity. Prove that there do not exist integers
a
,
b
,
c
,
d
,
k
a, b, c, d, k
a
,
b
,
c
,
d
,
k
with
k
>
1
k > 1
k
>
1
such that
(
a
+
b
ω
+
c
ω
2
+
d
ω
3
)
k
=
1
+
ω
.
(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.
(
a
+
bω
+
c
ω
2
+
d
ω
3
)
k
=
1
+
ω
.
Carl Lian
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