Prime Difference Function


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

 lim inf_(n->infty)(p_(n+1)-p_n)/(lnp_n)=0

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

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

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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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. 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.", 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."

Referenced on Wolfram|Alpha

Prime Difference Function

Cite this as:

Weisstein, Eric W. "Prime Difference Function." From MathWorld--A Wolfram Web Resource.

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