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
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
(Montgomery 2003). Unfortunately, this proof turned out to be flawed.
is called a jumping champion if is the most frequently occurring difference between consecutive
(Odlyzko et al.).
See alsoAndrica's Conjecture
, Cramér Conjecture
, Good Prime
, Pólya Conjecture
, Prime Gaps
, Twin Peaks
Explore with Wolfram|Alpha
ReferencesBombieri, 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. New York: Springer-Verlag, pp. 19-23
and 26-27, 1994.Havil, J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 114-115,
2003.Montgomery, H. "Small Gaps Between Primes." 13 Mar 2003.
A.; Rubinstein, M.; and Wolf, M. "Jumping Champions." http://www.research.att.com/~amo/doc/recent.html.Riesel,
H. "Difference Between Consecutive Primes." Prime
Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser,
p. 9, 1994.Sloane, N. J. A. Sequence A001223/M0296
in "The On-Line Encyclopedia of Integer Sequences."
on Wolfram|AlphaPrime Difference Function
Cite this as:
Weisstein, Eric W. "Prime Difference Function."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeDifferenceFunction.html