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THE JUMPING CHAMPION CONJECTURE

Part of:Additive number theory; partitionsMultiplicative number theory

Published online by Cambridge University Press:  04 May 2015

Daniel A. Goldston and
Andrew H. Ledoan
Daniel A. Goldston
Affiliation:
Department of Mathematics, San José State University, 315 MacQuarrie Hall, One Washington Square, San José, CA 95192-0103, U.S.A. email daniel.goldston@sjsu.edu
Andrew H. Ledoan
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, 415 EMCS Building (Mail Stop 6956), 615 McCallie Avenue, Chattanooga, TN 37403-2598, U.S.A. email andrew-ledoan@utc.edu

Abstract

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An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally, several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. In 1999, Odlyzko et al provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $2,6,30,210,2310,\ldots \,$. In this paper, we prove that an appropriate form of the Hardy–Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$.

MSC classification

Primary:11N05: Distribution of primes
Secondary:11P32: Goldbach-type theorems; other additive questions involving primes11N36: Applications of sieve methods

Information

Type
Research Article
Information
Mathematika , Volume 61 , Issue 3 , September 2015, pp. 719 - 740
Copyright
Copyright © University College London 2015 

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