An integer is called a Jumping Champion if is the most frequently occurring difference between consecutive
primes for some (Odlyzko *et al. *). This term was coined by J. H. Conway in 1993. There are occasionally several
jumping champions in a range. Odlyzko *et al. *give a table of jumping champions for , consisting mainly of 2, 4,
and 6. 6 is the jumping champion up to about
, at which point 30 dominates. At
, 210 becomes champion, and subsequent Primorials are conjectured to take over at larger and
larger . Erdös and Straus (1980) proved that the jumping champions tend to infinity under the assumption of a
quantitative form of the -tuples conjecture.

**References**

Erdös, P.; and Straus, E. G. ``Remarks on the Differences Between Consecutive Primes.'' *Elem. Math.* **35**, 115-118, 1980.

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, 1994.

Nelson, H. ``Problem 654.'' *J. Recr. Math.* **11**, 231, 1978-1979.

Odlyzko, A.; Rubinstein, M.; and Wolf, M. ``Jumping Champions.'' http://www.research.att.com/~amo/doc/recent.html.

© 1996-9

1999-05-25