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I wonder if the following is known:

let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where $x, y$ were positive integers and the ratio $y/x$ would approach zero?

Thank you!

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Yes, this can be proved similarly as Dirichlet's theorem on arithmetic progressions. More precisely, using $L(1,\chi)\neq 0$ for all the nontrivial unramified Grössencharacters of $\mathbb{Q}(\sqrt{-n})$ it can be proved that the prime ideals of this number field are equidistributed in the class group and, within each ideal class, they are also equidistributed in "angle". In particular, for any $\epsilon>0$, there are infinitely many principal prime ideals $(x+y\sqrt{-n})$ such that $0<y/x<\epsilon$. As $x^2+ny^2$ is a rational prime here, this implies the affirmative answer to your question. For details I recommend Neukirch: Algebraische Zahlentheorie (English translation is available).

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