Romanov's Theorem in Number Fields

TL;DR

This paper extends Romanov's theorem to number fields, providing explicit bounds on the proportion of algebraic integers that are sums of a prime and a fixed power, with specific results for Gaussian integers.

Contribution

It determines explicit lower bounds for the density of such representations in number fields and improves bounds for Gaussian integers, also constructing special progressions with few representations.

Findings

Explicit lower bounds for algebraic integers as sums of prime and power.

Improved lower bounds for Gaussian integers with such representations.

Construction of progressions with almost no such representations.

Abstract

Romanov proved that a positive proportion of the integers have a representation as a sum of a prime and a power of an arbitrary fixed positive integer. Rieger proved the analogous result for number fields. We will determine an explicit lower bound for the proportion of algebraic integers in a given number field, which are sums of a power of a fixed non-unit and a prime. Furthermore, we give an improved lower bound for the lower density of Gaussian integers that have a representation as a sum of a Gaussian prime and a power of $1+i$. Finally, similar to Erd\H{o}s, we construct an explicit arithmetic progression of Gaussian integers with odd norm such that almost all elements of this progression do not have a representation as the sum of a prime and a power of $1+i$.