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Distinct positive integers

A

and

B

are given

.

Prove that there exist infinitely many positive integers that can be represented both as

x_{1}^2+Ay_{1}^2

for some positive coprime integers

x_{1}

and

y_{1},

and as

x_{2}^2+By_{2}^2

for some positive coprime integers

x_{2}

and

y_{2}.

(Golovanov A.S.)

Problem Statement