The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEIS
A001223). Rankin has shown that

(2)

for infinitely many and for some constant (Guy 1994). At a March 2003 meeting on elementary and analytic
number in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof
that

(3)

(Montgomery 2003). Unfortunately, this proof turned out to be flawed.

An integer
is called a jumping champion if is the most frequently occurring difference between consecutive
primes
for some
(Odlyzko et al.).