The prime difference function is defined by
(1)
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
(3)
(Montgomery 2003). Unfortunately, this proof turned out to be flawed.
An integer jumping champion if et al. 1999).
See also Andrica's Conjecture ,
Cramér Conjecture ,
Gilbreath's
Conjecture ,
Good Prime ,
Jumping
Champion ,
Pólya Conjecture ,
Prime
Distance ,
Prime Gaps ,
Shanks'
Conjecture ,
Twin Peaks
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References Bombieri, E. and Davenport, H. "Small Differences between Prime Numbers." Proc. Roy. Soc. A 293 , 1-18, 1966. Erdős,
P.; and Straus, E. G. "Remarks on the Differences between Consecutive Primes."
Elem. Math. 35 , 115-118, 1980. Guy, R. K. "Gaps
between Primes. Twin Primes" and "Increasing and Decreasing Gaps."
§A8 and A11 in Unsolved
Problems in Number Theory, 2nd ed. Havil, J. Gamma:
Exploring Euler's Constant. Montgomery, H. "Small Gaps Between Primes." 13 Mar 2003.
https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0303&L=nmbrthry&P=1323 . Odlyzko,
A.; Rubinstein, M.; and Wolf, M. "Jumping Champions." Experiment. Math. 8 ,
107-118, 1999. https://doi.org/10.1080/10586458.1999.10504393 . Riesel,
H. "Difference Between Consecutive Primes." Prime
Numbers and Computer Methods for Factorization, 2nd ed. Sloane, N. J. A. Sequence A001223 /M0296
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Prime Difference Function
Cite this as:
Weisstein, Eric W. "Prime Difference Function."
From MathWorld https://mathworld.wolfram.com/PrimeDifferenceFunction.html
Subject classifications
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
is called a jumping champion if 
is the most frequently occurring difference between consecutive
primes 
for some 
(Odlyzko et al. 1999).
