The jumping champion conjecture

TL;DR

This paper proves that under certain conjectures, all large jumping champions are primorials and vice versa, linking prime gaps to the structure of primorial numbers.

Contribution

It establishes a conditional proof connecting large jumping champions to primorials based on the Hardy-Littlewood prime $k$-tuple conjecture.

Findings

Large jumping champions are primorials under the conjecture.

All sufficiently large primorials are jumping champions.

The results depend on the prime $k$-tuple conjecture.

Abstract

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 $x$. For the $n$th prime $p_{n}$, the $n$th primorial $p_{n}^{\sharp}$ is defined as the product of the first $n$ primes. 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 $p_{1}^{\sharp}, p_{2}^{\sharp}, p_{3}^{\sharp}, p_{4}^{\sharp}, p_{5}^{\sharp}, ...$, that is, $2, 6, 30, 210, 2310, ....$ 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 $x$.