Year: 2015
Type: preprint
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 Erdős, 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 $1+i$.
Source: arXiv (Cornell University)
Authors: Madritsch, Manfred G., Planitzer, Stefan
Institutions: Institut Élie Cartan de Lorraine, Université de Lorraine
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2281209999