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Prime Quadruplet


A prime constellation of four successive primes with minimal distance (p,p+2,p+6,p+8). The term was coined by Paul Stäckel (1892-1919; Tietze 1965, p. 19). The quadruplet (2, 3, 5, 7) has smaller minimal distance, but it is an exceptional special case. With the exception of (5, 7, 11, 13), a prime quadruple must be of the form (30n+11, 30n+13, 30n+17, 30n+19). The first few values of n which give prime quadruples are n=0, 3, 6, 27, 49, 62, 69, 108, 115, ... (OEIS A014561), and the first few values of p are 5 (the exceptional case), 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, ... (OEIS A007530). The number of prime quadruplets with largest member less than 10^1, 10^2, ..., are 1, 2, 5, 12, 38, 166, 899, 4768, ... (OEIS A050258; Nicely 1999).

The asymptotic formula for the frequency of prime quadruples is analogous to that for other prime constellations,

P_x(p,p+2,p+6,p+8)∼(27)/2product_(p>=5)(p^3(p-4))/((p-1)^4)int_2^x(dx)/((lnx)^4)
(1)
=4.151180864int_2^x(dx)/((lnx)^4),
(2)

where c=4.15118... (OEIS A061642) is the Hardy-Littlewood constant for prime quadruplets.

Roonguthai found the large prime quadruplets with

p=10^(99)+349781731
(3)
p=10^(199)+21156403891
(4)
p=10^(299)+140159459341
(5)
p=10^(399)+34993836001
(6)
p=10^(499)+883750143961
(7)
p=10^(599)+1394283756151
(8)
p=10^(699)+547634621251.
(9)

Forbes found the large quadruplet with

 p=76912895956636885(2^(3279)-2^(1093))-6·2^(1093)-7.
(10)

See also

k-Tuple Conjecture, Prime Arithmetic Progression, Prime Constellation, Prime Triplet, Sexy Primes, Twin Primes

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References

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. New York: Oxford University Press, 1979.Forbes, T. "Prime k-tuplets." http://anthony.d.forbes.googlepages.com/ktuplets.htm.Forbes, T. "Large Prime Quadruplets." 17 Sep 1998. http://listserv.nodak.edu/scripts/wa.exe?A2=ind9809&L=nmbrthry&P=992.Nicely, T. R. "Enumeration to 1.6×10^(15) of the Prime Quadruplets." Submitted to Math. Comput. http://www.trnicely.net/quads/quads.html.Rademacher, H. Lectures on Elementary Number Theory. New York: Blaisdell, 1964.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 61-62, 1994.Finch, S. R. "Hardy-Littlewood Constants." §2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 84-94, 2003.Sloane, N. J. A. Sequences A007530/M3816, A014561, A050258, and A061642 in "The On-Line Encyclopedia of Integer Sequences."Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, p. 19, 1965.

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Prime Quadruplet

Cite this as:

Weisstein, Eric W. "Prime Quadruplet." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PrimeQuadruplet.html

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