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Number Theory Problem
Source: SRMC 2022 P2
5/23/2022
Distinct positive integers
A
A
A
and
B
B
B
are given
.
.
.
Prove that there exist infinitely many positive integers that can be represented both as
x
1
2
+
A
y
1
2
x_{1}^2+Ay_{1}^2
x
1
2
+
A
y
1
2
for some positive coprime integers
x
1
x_{1}
x
1
and
y
1
,
y_{1},
y
1
,
and as
x
2
2
+
B
y
2
2
x_{2}^2+By_{2}^2
x
2
2
+
B
y
2
2
for some positive coprime integers
x
2
x_{2}
x
2
and
y
2
.
y_{2}.
y
2
.
(Golovanov A.S.)
number theory