Jumping Champion

An integer $n$ is called a Jumping Champion if $n$ is the most frequently occurring difference between consecutive primes $n\leq N$ for some $N$ (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 $n\leq 1000$, consisting mainly of 2, 4, and 6. 6 is the jumping champion up to about $n\approx 1.74\times 10^{35}$, at which point 30 dominates. At $n\approx 10^{425}$, 210 becomes champion, and subsequent Primorials are conjectured to take over at larger and larger $n$. Erdös and Straus (1980) proved that the jumping champions tend to infinity under the assumption of a quantitative form of the $k$-tuples conjecture.

See also Prime Difference Function, Prime Gaps, Prime Number, Primorial

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.