Polignac's conjecture

In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states:

For n = 2, it is the twin prime conjecture. For n = 4, it says there are infinitely many cousin primes (p, p + 4). For n = 6, it says there are infinitely many sexy primes (p, p + 6) with no prime between p and p + 6.

Dickson's conjecture generalizes Polignac's conjecture to cover all prime constellations.

Let ${\displaystyle \pi _{n}(x)}$ for even n be the number of prime gaps of size n below x.

The first Hardy–Littlewood conjecture says the asymptotic density is of form

where C_n is a function of n, and ${\displaystyle \sim }$ means that the quotient of two expressions tends to 1 as x approaches infinity.

C₂ is the twin prime constant

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