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The prime ktuple conjecture states that every admissible pattern for a
prime constellation occurs infinitely often and that the number of occurrences
of a prime constellation of length k is (infinitely often) greater than
a constant times x/(log x)^{k}.
Consider the cases k=2, 3, and 4. Here Hardy and Littlewood
heuristically estimated the number of each pattern less
than x is
See Also: PrimeConstellation, PrimeNumberThm, DicksonsConjecture Related pages (outside of this work) References:
 Guy94 (A9)
 R. K. Guy, Unsolved problems in number theory, SpringerVerlag, New York, NY, 1994. ISBN 0387942890. MR 96e:11002 [An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]
 HL23
 G. H. Hardy and J. E. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes," Acta Math., 44 (1923) 170. Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561630, Clarendon Press, Oxford, 1966.
 Ribenboim95
 P. Ribenboim, The new book of prime number records, 3rd edition, SpringerVerlag, New York, NY, 1995. pp. xxiv+541, ISBN 0387944575. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventyfive pages.]
 Riesel94 (Chapter Three)
 H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics Vol, 126, Birkhäuser Boston, Boston, MA, 1994. ISBN 0817637435. MR 95h:11142 [An excellent reference for those who want to start to program some of these algorithms. Code is provided in Pascal. Previous edition was vol. 57, 1985.]
