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).
 |
(5)
|
is a special case of the 
-tuple conjecture with 
, 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 
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.
