A binary problem
type: Item
instance of: scholarly article
title: A binary problem
zbMATH Open document ID: 0902.11004
DOI: 10.1007/BF02305438
published in: Mathematical Notes
publication date: 1998-08-04T00:00:00Z
review text: Let \(n=\sum_{k=0}^{\infty} \varepsilon_k 2^k\) be the binary decomposition of the positive integer \(n\), where \(\varepsilon_k \in \{0,1\}, k=0,1,2,\dots\), and let \(\mathbb{N}_i\) denote the set of positive integers \(n\) such that \(\sum_{k=0}^{\infty} \varepsilon_k\equiv i\) (mod \(2\)), where \(i\in \{0,1\}\). Furthermore, denote by \(F_{i,j}\) the number of solutions of the equation \(n-m=1\), where \(n,m\leq x, n\in \mathbb{N}_i, m\in \mathbb{N}_j, i,j\in \{0,1\}\). The author shows that the following asymptotic formulae hold for \(i,j\in \{0,1\}\): \[ F_{i,j}=C_{i,j}x + O(\log x), \] where \(C_{0,0}= C_{1,1}=1/6\) and \(C_{1,0}= C_{0,1}=1/3\).
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 1158943
zbMATH Keywords: binary decomposition
zbMATH Keywords: asymptotic formula
author: K. M. Èminyan
reviewed by: László Tóth
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf02305438
OpenAlex ID: W2086113042
description: scientific article; zbMATH DE number 1158943