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Polish MO Finals
1965 Polish MO Finals
2
2
Part of
1965 Polish MO Finals
Problems
(1)
x_1^n + x_2^n and x_1^{n+1} + x_2^{n+1} are integer and coprime.
Source: Polish MO Finals 1965 p2
8/30/2024
Prove that if the numbers
x
1
x_1
x
1
and
x
2
x_2
x
2
are roots of the equation
x
2
+
p
x
−
1
=
0
x^2 + px - 1 = 0
x
2
+
p
x
−
1
=
0
, where
p
p
p
is an odd number, then for every natural
n
n
n
number
x
1
n
+
x
2
n
x_1^n + x_2^n
x
1
n
+
x
2
n
and
x
1
n
+
1
+
x
2
n
+
1
x_1^{n+1} + x_2^{n+1}
x
1
n
+
1
+
x
2
n
+
1
are integer and coprime.
algebra
number theory
coprime
Integer