Primes of the form \pm a^2\pm qb^2

TL;DR

This paper explores the representation of primes using quadratic forms of the type ^2^2, providing a comprehensive list for q  20, and introduces new relationships between different representations, combining elementary and novel methods.

Contribution

It offers a comprehensive classification of primes represented by ^2^2 forms for q  20 and introduces new relationships between these representations.

Findings

List of primes represented by ^2^2 forms for q  20

Elementary methods applied to establish some results

New relationships between different quadratic representations

Abstract

Representations of primes by simple quadratic forms, such as $\pm a^2\pm qb^2$, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gauss and many others. We are interested in a comprehensive list of such results, for $q\le 20$. Some of the results can be established with elementary methods and we put them at work on some instances. We are introducing new relationships between various representations.