Primes of the form $x^2+dy^2$ with $x\equiv 0\pmod{N}$ or $y\equiv 0\pmod{N}$

TL;DR

This paper characterizes primes of the form x^2 + dy^2 where either x or y is divisible by N, for positive integers N and square-free d, extending understanding of prime representations in quadratic forms.

Contribution

It provides a new characterization of primes of the form x^2 + dy^2 with divisibility conditions on x or y, for general N and square-free d.

Findings

Identifies conditions for primes of the form x^2 + dy^2 with x or y divisible by N

Extends classical results on primes of quadratic forms to divisibility constraints

Provides a framework for analyzing primes in specific quadratic form classes

Abstract

In this paper we charatcterize primes of the form $x^2+dy^2$ with $x\equiv 0\pmod{N}$ or $y\equiv 0\pmod{N}$ for positive integer $N$ and $d$ with $d$ being square free.