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CIIM
2023 CIIM
5
5
Part of
2023 CIIM
Problems
(1)
Find all polynomial with modulus 1 root
Source: CIIM 2023 - Problem 5
9/19/2023
Given a positive integer
k
>
1
k > 1
k
>
1
, find all positive integers
n
n
n
such that the polynomial
P
(
z
)
=
z
n
+
∑
j
=
0
2
k
−
2
z
j
=
1
+
z
+
z
2
+
⋯
+
z
2
k
−
2
+
z
n
P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n
P
(
z
)
=
z
n
+
j
=
0
∑
2
k
−
2
z
j
=
1
+
z
+
z
2
+
⋯
+
z
2
k
−
2
+
z
n
has a complex root
w
w
w
such that
∣
w
∣
=
1
|w| = 1
∣
w
∣
=
1
.
algebra
polynomial
complex numbers
modulus