The jumping champion conjecture - Goldston, D A; Ledoan, A H (arXiv.org, 2012)
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 \(x\).