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Prime Difference Function

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PrimeDifferenceFunction

The prime difference function d_n is the difference between consecutive primes,

d_n=p_(n+1)-p_n,
(1)

where p_n is the nth prime and n is a positive integer. 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

d_n>(clnnlnlnnlnlnlnlnn)/((lnlnlnn)^2)
(2)

for infinitely many n and for some positive 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

liminf_(n->infty)(p_(n+1)-p_n)/(lnp_n)=0
(3)

(Montgomery 2003). Unfortunately, this proof turned out to be flawed. The equality was subsequently proved unconditionally by Goldston et al. (2009, p. 819).

A positive integer d is a jumping champion if it is a most frequently occurring difference between consecutive primes not exceeding some bound N (Odlyzko 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.Goldston, D. A.; Pintz, J.; and Yildirim, C. Y. "Primes in Tuples I." Ann. Math. 170, 819-862, 2009. https://doi.org/10.4007/annals.2009.170.819.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." Mar. 13, 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. Boston, MA: Birkhäuser, p. 9, 1994.Sloane, N. J. A. Sequence A001223/M0296 in "The On-Line Encyclopedia of Integer Sequences."

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Prime Difference Function

Cite this as:

Weisstein, Eric W. "Prime Difference Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimeDifferenceFunction.html

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