Representing Primes as the Form $x^2+ny^2$ in Some Imaginary Quadratic Fields

TL;DR

This paper establishes criteria for when primes can be expressed as $x^2+ny^2$ in certain imaginary quadratic fields, simplifying the understanding of such representations in specific cases.

Contribution

It provides explicit solvability criteria for the equation $p=x^2+ny^2$ over some imaginary quadratic fields, enhancing the theoretical framework for prime representations.

Findings

Criteria for solvability of $p=x^2+ny^2$ in specific imaginary quadratic fields

Simplified conditions in special cases

Improved understanding of prime representations in quadratic fields

Abstract

We give criteria of the solvability of the diophantine equation $p=x^2+ny^2$ over some imaginary quadratic fields where $p$ is a prime element. The criteria becomes quite simple in special cases.