Very large gaps between consecutive primes
type: Item
instance of: scholarly article
title: Very large gaps between consecutive primes
zbMATH Open document ID: 0870.11056
author: János Pintz
published in: Journal of Number Theory
publication date: 1997-09-11T00:00:00Z
review text: Let \(G(x)=\max p_{n+1}-p_n\) for primes \(p_n\leq x\). Then it is shown that \[ G(x)\geq (2e^\gamma+o(1)) \frac{(\log x)(\log\log x)(\log\log\log\log x)}{(\log \log\log x)^2}. \] This improves results of \textit{R. A. Rankin} [Proc. Edinb. Math. Soc., II. Ser. 13, 331-332 (1963; Zbl 0121.04705)] (with constant \(e^\gamma\)) and \textit{H. Maier} and \textit{C. Pomerance} [Trans. Am. Math. Soc. 322, 201-237 (1990; Zbl 0706.11052)] (with constant \((1.312\dots)e^\gamma)\). In both these papers one arrives at a set \(R\) of integers that must be covered by congruences \(a_p\pmod p\) for suitable primes \(p\). The numbers \(a_p\) are free to be chosen. Rankin essentially uses each prime \(p\) to cover one element of \(R\). Maier and Pomerance show that one can often delete two elements of \(R\) with the same prime \(p\). The present paper shows that one can almost always delete two elements of \(R\) for each prime. The result and its proof are phrased in terms of colourings of graphs. The existence of a suitable colouring is established by a probabilistic argument.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11N05
zbMATH DE Number: 1011055
zbMATH Keywords: large gaps between consecutive primes
zbMATH Keywords: colourings of graphs
MaRDI profile type: Publication
OpenAlex ID: W2095527188
Wikidata QID: Q60043294
cites work: Unusually Large Gaps Between Consecutive Primes
cites work: The Difference between Consecutive Prime Numbers
cites work: The Difference between Consecutive Prime Numbers V
DOI: 10.1006/JNTH.1997.2081
description: scientific article; zbMATH DE number 1011055