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

TL;DR

This paper classifies pairs of distinct odd primes p and q for which the Diophantine equation pq=x^2+ny^2 has solutions, focusing on cases n=5 and 14, expanding understanding of prime representations in quadratic forms.

Contribution

It provides a complete classification of prime pairs p and q satisfying pq=x^2+ny^2 for specific n values, advancing knowledge on prime representations in quadratic equations.

Findings

Classified all prime pairs p and q for n=5 and 14.

Identified conditions under which pq=x^2+ny^2 has solutions.

Enhanced understanding of prime representations in quadratic forms.

Abstract

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 $14$.