On a conjecture of Graham
type: Item
instance of: scholarly article
title: On a conjecture of Graham
zbMATH Open document ID: 0629.10003
DOI: 10.1016/0022-314X(87)90048-5
author: Alexandru Zaharescu
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: A well-known problem due to \textit{R. L. Graham} [Am. Math. Mon. 77, 775 (1970)] is the following: is it true that, for every chain \(0<a_ 1<...<a_ n\) of integers, there exist i,j such that \(\max_{i,j}(a_ i/(a_ i,a_ j))\geq n ?\) Let p(n) denote the largest prime number \(<2n\), and put \(f(n)=2n-p(n)\). The author shows that the answer to Graham's question is in the affirmative for all n for which \(f(n)<\sqrt{n}\). He then refines his argument to make use of Huxley's result \(f(n)\leq n^{7/12+\epsilon}\) for all sufficiently large n to confirm that Graham's conjecture is true when n is sufficiently big. [See \textit{M. Szegedy}, Combinatorica 6, 67-71 (1986; Zbl 0593.10002) for a proof of Graham's conjecture.]
Mathematics Subject Classification ID: 11A05
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 4023365
zbMATH Keywords: gaps between primes
zbMATH Keywords: greatest common divisor
zbMATH Keywords: Graham's conjecture
Wikidata QID: Q123270714
MaRDI profile type: Publication
cites work: Q5661970
description: scientific article; zbMATH DE number 4023365