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Divisibility problem by a^2+ab+b^2
Divisibility problem by a^2+ab+b^2
Source:
September 29, 2010
number theory
Divisibility
IMO Shortlist
IMO Longlist
Problem Statement
(
H
U
N
1
)
(HUN 1)
(
H
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)
Let
a
a
a
and
b
b
b
be arbitrary integers. Prove that if
k
k
k
is an integer not divisible by
3
3
3
, then
(
a
+
b
)
2
k
+
a
2
k
+
b
2
k
(a + b)^{2k}+ a^{2k} +b^{2k}
(
a
+
b
)
2
k
+
a
2
k
+
b
2
k
is divisible by
a
2
+
a
b
+
b
2
a^2 +ab+ b^2
a
2
+
ab
+
b
2
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