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Prime Quadruplet

A prime constellation of four successive primes with minimal distance . 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 (, , , ). The first few values of which give prime quadruples are , 3, 6, 27, 49, 62, 69, 108, 115, ... (Sloane's A014561), and the first few values of are 5 (the exceptional case), 11, 101, 191, 821, 1481, 1871, 2081, 3251, 3461, ... (Sloane's A007530). The number of prime quadruplets with largest member less than , , ..., are 1, 2, 5, 12, 38, 166, 899, 4768, ... (Sloane's A050258; Nicely 1999).

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

			(1)
			(2)

where (Sloane's A061642) is the Hardy-Littlewood constant for prime quadruplets.

Roonguthai found the large prime quadruplets with

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)

(Roonguthai). Forbes found the large quadruplet with

(10)

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 -tuplets." http://www.ltkz.demon.co.uk/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 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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Weisstein, Eric W. "Prime Quadruplet." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PrimeQuadruplet.html