Romanov's Theorem in Number Fields

Manfred G. Madritsch; Stefan Planitzer

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arXiv · 1512.04869

Romanov's Theorem in Number Fields

Manfred G. Madritsch·Stefan Planitzer

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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 $1+i$.

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BibTeX RIS

Manfred G. Madritsch, Stefan Planitzer. 2015-12-15. Romanov's Theorem in Number Fields. https://arxiv.org/abs/1512.04869

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