Threefield identities and simultaneous representations of primes by binary quadratic forms

TL;DR

This paper explores new theorems on simultaneous prime representations by binary quadratic forms, extending Kaplansky's work through threefield identities and revealing additional results beyond Brink's list.

Contribution

It introduces new theorems on prime representations by quadratic forms using threefield identities, expanding the known results in the field.

Findings

Identifies at least two new theorems on prime representations

Connects threefield identities with binary quadratic form results

Extends Kaplansky's and Brink's theorems with new findings

Abstract

Kaplansky [2003] proved a theorem on the simultaneous representation of a prime $p$ by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink's list. We also show how such theorems are related to results of Muskat on binary quadratic forms