Constellations in prime elements of number fields

TL;DR

This paper proves the existence of arbitrarily shaped constellations of prime elements in number fields, extending Green-Tao and Tao's results to broader algebraic settings and quadratic forms.

Contribution

It introduces new theorems demonstrating the existence of prime constellations in number fields and quadratic forms, generalizing previous prime pattern results.

Findings

Existence of arbitrarily shaped prime constellations in number fields

Extension of Green-Tao and Tao's theorems to algebraic number theory

Prime representations of binary quadratic forms with prime values

Abstract

Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.