On the densities of some sequences
type: Item
instance of: scholarly article
title: On the densities of some sequences
zbMATH Open document ID: 0769.11005
DOI: 10.1016/0022-314X(92)90110-B
author: Arulappah Eswarathasan
author: David Lubell
published in: Journal of Number Theory
publication date: 1993-05-16T00:00:00Z
review text: In 1934, \textit{N. P. Romanoff} [Math. Ann. 109, 668-678 (1934; Zbl 0009.00801)] proved that the integers of the form \(p+2^ m\), with \(p\) prime, have positive density in \(\mathbb{Z}\). In this paper, the authors generalize this result to Gaussian integers: If \(\Lambda_ 1\) and \(\Lambda_ 2\) are fixed Gaussian integers with \(\Lambda^ 4_ 2=1\) and \(\left| \Lambda_ 1-\sqrt{\Lambda^ 2_ 1+4\Lambda_ 2}\right|\neq\left|\Lambda_ 1+\sqrt{\Lambda^ 2_ 1 +4\Lambda_ 2}\right|\) then \(\{\pi+\varphi_ n\}\) has positive density in \(\mathbb{Z}[i]\), where \(\pi\) varies over the Gaussian primes and \(\varphi_ n\) is defined by \(\varphi_ 0=0\), \(\varphi_ 1=1\), \(\varphi_{n+2}=\Lambda_ 1\varphi_{n+1}+\Lambda_ 2\varphi_ n\). Related questions are also discussed.
Mathematics Subject Classification ID: 11B05
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 166049
zbMATH Keywords: Gaussian integers
zbMATH Keywords: positive density
reviewed by: M. Nair
MaRDI profile type: Publication
cites work: On Pseudo-Lucas Numbers of the form 2s2
cites work: Q5800860
description: scientific article; zbMATH DE number 166049