k-Tuple Conjecture
المؤلف:
Brent, R. P.
المصدر:
"The Distribution of Small Gaps Between Successive Primes." Math. Comput. 28
الجزء والصفحة:
...
6-9-2020
2406
k-Tuple Conjecture
The first of the Hardy-Littlewood conjectures. The 
-tuple conjecture states that the asymptotic number of prime constellations can be computed explicitly. In particular, unless there is a trivial divisibility condition that stops 
, 
, ..., 
from consisting of primes infinitely often, then such prime constellations 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 et al. 1999). If 
, then this becomes
 |
(4)
|
This conjecture is generally believed to be true, but has not been proven (Odlyzko et al. 1999).
The twin prime conjecture
 |
(5)
|
is a special case of the 
-tuple conjecture with ![S=<span style=]()
{0,2}" src="https://mathworld.wolfram.com/images/equations/k-TupleConjecture/Inline19.gif" style="height:15px; width:57px" />, where 
is known as the twin primes constant.
The following special case of the conjecture is sometimes known as the prime patterns conjecture. Let 
be a finite set of integers. Then it is conjectured that there exist infinitely many 
for which ![<span style=]()
{k+s:s in S}" src="https://mathworld.wolfram.com/images/equations/k-TupleConjecture/Inline23.gif" style="height:15px; width:77px" /> are all prime iff 
does not include all the residues of any prime. This conjecture also implies that there are arbitrarily long arithmetic progressions of 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, 1923.
Odlyzko, A.; Rubinstein, M.; and Wolf, M. "Jumping Champions." Experiment. Math. 8, 107-118, 1999.
Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 66-68, 1994.
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