Prime Quadruplet
A Prime Constellation of four successive Primes with minimal distance 
. 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 quadruplemust 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, .... The asymptotic Formula for the frequency of prime quadruplesis analogous to that for other Prime Constellations,
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where
is the Hardy-Littlewood constant for prime quadruplets.Roonguthai found the large prime quadruplets with  | |  | |
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(Roonguthai).See also Prime Arithmetic Progression, Prime Constellation, Prime k-Tuples Conjecture, Sexy Primes, Twin Primes
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 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. Roonguthai, W. ``Large Prime Quadruplets.'' http://www.mathsoft.com/asolve/constant/hrdyltl/roonguth.html. Sloane, N. J. A. Sequence A014561in ``The On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
-tuplets.'' http://www.ltkz.demon.co.uk/ktuplets.htm.