On the Diophantine equation $pq=x^2+ny^2$

Ja Kyung Koo, Dong Hwa Shin

Published 2014-04-03, updated 2014-04-17Version 2

Let $n$ be a positive integer. We discuss pairs of distinct odd primes $p$ and $q$ not dividing $n$ for which the Diophantine equations $pq=x^2+ny^2$ have integer solutions in $x$ and $y$. As its examples we classify all such pairs of $p$ and $q$ when $n=5$ and $14$.