Polignac's conjecture

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

For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.^[1]

The conjecture has not yet been proven or disproven for a given value of n. In 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size n for some value of n < 70,000,000.^[2]^[3] Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600.^[4] As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki, n has been reduced to 246.^[5] Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that n has been reduced to 12 and 6, respectively.^[6]

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.

Conjectured density

Let $\pi _{n}(x)$ 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

\pi _{n}(x)\sim 2C_{n}{\frac {x}{(\ln x)^{2}}}\sim 2C_{n}\int _{2}^{x}{dt \over (\ln t)^{2}}

where C_n is a function of n, and $\sim$ means that the quotient of two expressions tends to 1 as x approaches infinity.^[7]

C₂ is the twin prime constant

C_{2}=\prod _{p\geq 3}{\frac {p(p-2)}{(p-1)^{2}}}\approx 0.660161815846869573927812110014\dots

where the product extends over all prime numbers p ≥ 3.

C_n is C₂ multiplied by a number which depends on the odd prime factors q of n:

C_{n}=C_{2}\prod _{q|n}{\frac {q-1}{q-2}}.

For example, C₄ = C₂ and C₆ = 2C₂. Twin primes have the same conjectured density as cousin primes, and half that of sexy primes.

Note that each odd prime factor q of n increases the conjectured density compared to twin primes by a factor of ${\tfrac {q-1}{q-2}}$ . A heuristic argument follows. It relies on some unproven assumptions so the conclusion remains a conjecture. The chance of a random odd prime q dividing either a or a + 2 in a random "potential" twin prime pair is ${\tfrac {2}{q}}$ , since q divides 1 of the q numbers from a to a + q − 1. Now assume q divides n and consider a potential prime pair (a, a + n). q divides a + n if and only if q divides a, and the chance of that is ${\tfrac {1}{q}}$ . The chance of (a, a + n) being free from the factor q, divided by the chance that (a, a + 2) is free from q, then becomes ${\tfrac {q-1}{q}}$ divided by ${\tfrac {q-2}{q}}$ . This equals ${\tfrac {q-1}{q-2}}$ which transfers to the conjectured prime density. In the case of n = 6, the argument simplifies to: If a is a random number then 3 has chance 2/3 of dividing a or a + 2, but only chance 1/3 of dividing a and a + 6, so the latter pair is conjectured twice as likely to both be prime.