Hardy-Littlewood Conjectures
The first Hardy-Littlewood conjecture is called the k-Tuple Conjecture. It states that the asymptotic number ofPrime Constellations can be computed explicitly.
The second Hardy-Littlewood conjecture states that
for all
and 
, where 
is the Prime Counting Function. Although it is not obvious, Richards (1974)proved that this conjecture is incompatible with the first Hardy-Littlewood conjecture. See also Prime Constellation, Prime Counting Function
References
Richards, I. ``On the Incompatibility of Two Conjectures Concerning Primes.'' Bull. Amer. Math. Soc. 80, 419-438, 1974. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 61-62 and 68-69, 1994.
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