On Romanov's constant

Christian Elsholtz; Jan-Christoph Schlage-Puchta

doi:10.1007/s00209-017-1908-x

On Romanov's constant

Christian Elsholtz, Jan-Christoph Schlage-Puchta^*

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly one representation of the form p+2k has positive density. Previous results of this kind needed “at most 15” in place of “exactly one”. To achieve these results we introduce a new method. In particular we make use of uneven distribution of sums of a power of two and a reduced residue class.

Original language	English
Pages (from-to)	713-724
Journal	Mathematische Zeitschrift
Volume	288
DOIs	https://doi.org/10.1007/s00209-017-1908-x
Publication status	Published - 2018

ASJC Scopus subject areas

General Mathematics
Algebra and Number Theory

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Information, Communication & Computing

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10.1007/s00209-017-1908-x

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title = "On Romanov's constant",

abstract = "We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly one representation of the form p+2k has positive density. Previous results of this kind needed “at most 15” in place of “exactly one”. To achieve these results we introduce a new method. In particular we make use of uneven distribution of sums of a power of two and a reduced residue class.",

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AU - Schlage-Puchta, Jan-Christoph

PY - 2018

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AB - We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly one representation of the form p+2k has positive density. Previous results of this kind needed “at most 15” in place of “exactly one”. To achieve these results we introduce a new method. In particular we make use of uneven distribution of sums of a power of two and a reduced residue class.

KW - primes

KW - Romanov

U2 - 10.1007/s00209-017-1908-x

DO - 10.1007/s00209-017-1908-x

M3 - Article

VL - 288

SP - 713

EP - 724

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

ER -

On Romanov's constant

Abstract

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