Hardy-Littlewood Conjectures

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

$$\pi(x+y)-\pi(x)\leq\pi(y)$$

for all $x$ and $y$, where $\pi(x)$ 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.