  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 $x$.
