The largest integer expressible as a sum of reciprocal of integers
type: Item
instance of: scholarly article
title: The largest integer expressible as a sum of reciprocal of integers
zbMATH Open document ID: 0926.11020
author: H. Yokota
published in: Journal of Number Theory
publication date: 1999-11-22T00:00:00Z
review text: Given \(n\), denote by \(N(n)\) the set of integers \(k\), representable as a sum of at most \(n\) Egyptian fractions with denominators \(\leq n\), i. e. in the form \[ k = \sum_{\nu = 1}^n {\varepsilon_\nu \over \nu}, \quad \text{ where } \varepsilon_\nu \in \{0,1\}. \] The largest integer in this set is denoted by \( \displaystyle M(n) = \max_{a \in N(n)} a. \) The author is interested in precise estimates of \( M(n) \). He shows: There exist constants \( c_1, c_2, n \), such that for all \( n > n_0 \) the estimate \[ \log n + \gamma - 2 - {c_1 \over \log \log n} \leq M(n) \leq \log n + \gamma - {c_2 \over \log \log n} \] holds. \(\gamma\) is Euler's constant. For the complicated proof of this result in combinatorial number theory, the author makes use of Lemmata of several of his former papers [J. Number Theory 28, 272-282 (1988; Zbl 0637.10007); Can. Math. Bull. 33, 235-241 (1990; Zbl 0656.10012)].
reviewed by: Wolfgang Schwarz
Mathematics Subject Classification ID: 11D68
zbMATH DE Number: 1309361
zbMATH Keywords: Sums of Egyptian fractions
zbMATH Keywords: precise estimate of the largest integer representable as sum of reciprocals of different integers
zbMATH Keywords: combinatorial number theory
zbMATH Keywords: Erdős-Graham conjecture
MaRDI profile type: Publication
full work available at URL: jnth.1998.2359
OpenAlex ID: W2092052244
cites work: Q3872528
cites work: Approximate formulas for some functions of prime numbers
cites work: Q3123961
cites work: Denominators of Egyptian fractions
cites work: Length and denominators of Egyptian fractions. II
cites work: On Number of Integers Representable as Sums of Unit Fractions
cites work: On number of integers representable as a sum of unit fractions. II
DOI: 10.1006/JNTH.1998.2359
description: scientific article; zbMATH DE number 1309361