  The most common difference that occurs among the consecutive primes less than
or equal to $x$ is called a jumping champion. Occasionally there are ties.
Therefore there can be more than one jumping champion for a given $x$. In 1999
A. Odlyzko, M. Rubinstein, and M. Wolf provided heuristic and empirical
evidence in support of the conjecture that the numbers greater than 1 that are
jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,... As a step
towards proving this conjecture they introduced a second weaker conjecture that
any fixed prime $p$ divides all sufficiently large jumping champions. In this
paper we extend a method of P. Erd\H{o}s and E. G. Straus from 1980 to prove
that the second conjecture follows directly from the prime pair conjecture of
G. H. Hardy and J. E. Littlewood.

Number Theory