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Pré-Publication, Document De Travail Année : 2016

Romanov's theorem in number fields

Résumé

Romanov proved that a positive proportion of the integers has a representation as a sum of a prime and a power of an arbitrary fixed integer. We prove the analogous result for number fields. Furthermore we give an explicit 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 which do not have a representation of this type.

Dates et versions

hal-01279446 , version 1 (26-02-2016)

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Citer

Manfred G. Madritsch, Stefan Planitzer. Romanov's theorem in number fields. 2016. ⟨hal-01279446⟩
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