The first Hardy-Littlewood conjecture is called the *k*-Tuple Conjecture. It states that the asymptotic number of
Prime 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.

**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.

© 1996-9

1999-05-25