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Moldova Contests
Moldova Team Selection Test
2001 Moldova Team Selection Test
1
1
Part of
2001 Moldova Team Selection Test
Problems
(1)
Sum of integer powers belongs in A, while their sum doesn't
Source: Romanian IMO TST 2006, day 5, problem 1
5/23/2006
Let
n
n
n
be a positive integer of the form
4
k
+
1
4k+1
4
k
+
1
,
k
∈
N
k\in \mathbb N
k
∈
N
and
A
=
{
a
2
+
n
b
2
∣
a
,
b
∈
Z
}
A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}
A
=
{
a
2
+
n
b
2
∣
a
,
b
∈
Z
}
. Prove that there exist integers
x
,
y
x,y
x
,
y
such that
x
n
+
y
n
∈
A
x^n+y^n \in A
x
n
+
y
n
∈
A
and
x
+
y
∉
A
x+y \notin A
x
+
y
∈
/
A
.
number theory unsolved
number theory