“k-Tuple Conjecture”的概念、定义、习题解题方法及翻译-数学辞典

单词

k-Tuple Conjecture

释义

k-Tuple Conjecture

The first of the Hardy-Littlewood Conjectures. The -tuple conjecture states that the asymptotic number of PrimeConstellations can be computed explicitly. In particular, unless there is a trivial divisibilitycondition that stops , , ..., from consisting of Primes infinitely often, then such PrimeConstellations will occur with an asymptotic density which is computable in terms of , ...,. Let , then the -tuple conjecture predicts that the number of Primes such that, , ..., are all Prime is

(1)

where

(2)

the product is over Odd Primes

, and

(3)

denotes the number of distinct residues of 0,

, ...,

(mod

) (Halberstam and Richert 1974, Odlyzko). If

,then this becomes

(4)

This conjecture is generally believed to be true, but has not been proven (Odlyzko et al. ). The following special case of theconjecture is sometimes known as the Prime Patterns Conjecture. Let be a Finite set ofIntegers. Then it is conjectured that there exist infinitely many for which areall Prime Iff does not include all the Residues of anyPrime. The Twin Prime Conjecture is a special case of the prime patterns conjecture with . Thisconjecture also implies that there are arbitrarily long Arithmetic Progressions ofPrimes.

See also Arithmetic Progression, Dirichlet's Theorem, Hardy-Littlewood Conjectures, k-Tuple Conjecture, Prime Arithmetic Progression, Prime Constellation, PrimeQuadruplet, Prime Patterns Conjecture, Twin Prime Conjecture, Twin Primes

References

Brent, R. P. ``The Distribution of Small Gaps Between Successive Primes.'' Math. Comput. 28, 315-324, 1974.

Brent, R. P. ``Irregularities in the Distribution of Primes and Twin Primes.'' Math. Comput. 29, 43-56, 1975.

Halberstam, E. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974.

Hardy, G. H. and Littlewood, J. E. ``Some Problems of `Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes.'' Acta Math. 44, 1-70, 1922.

Odlyzko, A.; Rubinstein, M.; and Wolf, M. ``Jumping Champions.''

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 66-68, 1994.

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