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Baltic Way
2019 Baltic Way
18
18
Part of
2019 Baltic Way
Problems
(1)
At least one of b^2+b+1 and c^2+c+1 is composite.
Source: 2019 Baltic Way P18
11/18/2019
Let
a
,
b
a,b
a
,
b
, and
c
c
c
be odd positive integers such that
a
a
a
is not a perfect square and
a
2
+
a
+
1
=
3
(
b
2
+
b
+
1
)
(
c
2
+
c
+
1
)
.
a^2+a+1 = 3(b^2+b+1)(c^2+c+1).
a
2
+
a
+
1
=
3
(
b
2
+
b
+
1
)
(
c
2
+
c
+
1
)
.
Prove that at least one of the numbers
b
2
+
b
+
1
b^2+b+1
b
2
+
b
+
1
and
c
2
+
c
+
1
c^2+c+1
c
2
+
c
+
1
is composite.
number theory