Competition problem — Number theory

Number theory Difficulty 4.5 AIME Prove it Silk Road Mathematics Competition

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.)

Solution

Without loss of generality $A > B$ .
Choose an arbitrary prime $p > 2$ and let's find $x_1$ and $x_2$ so that
$x_1^2 + A(2p)^2 = x_2^2 + B(2p)^2,$
Hence, $x_2^2 - x_1^2 = 4Cp^2$ where $C = A - B$ . Set $x_1 = Cp^2 - 1$ and $x_2 = Cp^2 + 1$ . If $x_1$ and $x_2$ are both odd, then they are both coprime with $y = 2p$ , and we have $x_1^2 + Ay^2 = x_2^2 + By^2$ . If they are both even, then $\frac{x_1}{2}$ and $\frac{x_2}{2}$ are both coprime with $y = p$ , and we have $(\frac{x_1}{2})^2 + Ay^2 = (\frac{x_2}{2})^2 + By^2$ .
The number to which we have found two such forms will be not less than $p^2$ , thus proving there are infinitely many such numbers.

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