The Jumping Champion Conjecture

Details The Jumping Champion Conjecture The Jumping Champion Conjecture

2011-02-23

arxiv.org/abs/1102.4879v1

Mathematics

Number Theory

13 pages, 1 table

Scientific paper

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 and hence there can be more than one jumping champion for the same $x$. In 1999 Odlyzko, Rubinstein, and Wolf 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,... 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 $x$.

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