The first Hardy-Littlewood conjecture is called the k-tuple conjecture. It states that the asymptotic number of prime constellations can be computed explicitly. A particular case gives the so-called strong twin prime conjecture The second Hardy-Littlewood conjecture states that π(x + y) - π(x)<=π(y) for all x, y>=2, where π(x) is the prime counting function. The following table summarizes the first few values of π(x + y) - π(x) for integer y and x = 1, 2, .... The values of this function are plotted above.

Bouniakowsky conjecture | prime constellation | prime counting function | twin prime conjecture