# prime k-tuple conjecture

The prime k-tuple 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

(k=2) | |||

(k=3) | |||

(k=4) |

**See Also:** PrimeConstellation, PrimeNumberThm, DicksonsConjecture

**Related pages** (outside of this work)

**References:**

- Guy94 (A9)
R. K. Guy,Unsolved problems in number theory, Springer-Verlag, 1994. New York, NY, ISBN 0-387-94289-0.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. HardyandJ. E. Littlewood, "Some problems of `partitio numerorum' : III: on the expression of a number as a sum of primes,"Acta Math.,44(1923) 1-70. Reprinted in "Collected Papers of G. H. Hardy," Vol. I, pp. 561-630, Clarendon Press, Oxford, 1966.- Ribenboim95
P. Ribenboim,The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5.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 seventy-five pages.]- Riesel94 (Chapter Three)
H. Riesel,Prime numbers and computer methods for factorization, Progress in Mathematics Vol, 126, Birkhäuser Boston, 1994. Boston, MA, ISBN 0-8176-3743-5.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.]

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