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 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, .... The asymptotic Formula for the frequency of prime quadruples is analogous to that for other Prime Constellations,

where is the Hardy-Littlewood constant for prime quadruplets. Roonguthai found the large prime quadruplets with

(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 -tuplets.'' http://www.ltkz.demon.co.uk/ktuplets.htm.

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 A014561 in The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9 Eric W. Weisstein
1999-05-26