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2300
Roots of Unity Double Sum
Editorial
number theory
binomial theorem
roots of unity
complex numbers
2-adic valuation
Let
z
z
z
be a complex number with
z
1024
=
1
z^{1024} = 1
z
1024
=
1
and
z
512
≠
1
z^{512} \ne 1
z
512
=
1
. Compute
v
2
(
∑
k
=
0
1023
∑
m
=
0
1023
(
1
+
z
k
−
m
)
1024
)
,
v_2\!\left(\,\sum_{k=0}^{1023}\sum_{m=0}^{1023}(1+z^{k-m})^{1024}\right),
v
2
(
k
=
0
∑
1023
m
=
0
∑
1023
(
1
+
z
k
−
m
)
1024
)
,
where
v
2
(
n
)
v_2(n)
v
2
(
n
)
denotes the
2
2
2
-adic valuation of
n
n
n
.
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