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. A particular case gives the so-called strong twin prime conjecture

The second Hardy-Littlewood conjecture states that

for all , where is the prime counting function.

The following table summarizes the first few values of for integer and , 2, .... The values of this function are plotted above.

	OEIS	for , 2, ...
1	A080545	1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...
2	A090405	2, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, ...
3	A090406	2, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, ...

Although it is not obvious, Richards (1974) proved that the first and second conjectures are incompatible with each other.