2015/12/15 by Manfred G. Madritsch, Madritsch, Manfred G., Stefan Planitzer +1
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1512.04869
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.