In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states:[1] Although the conjecture has not yet been proven or disproven for any given value of n, in 2013 an important breakthrough was made by Yitang Zhang who proved that there are infinitely many prime gaps of size n for some value of n < 70,000,000.
[3][4] 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.
[5] As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki, n has been reduced to 246.
[6] 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.
The first Hardy–Littlewood conjecture says the asymptotic density is of form where Cn is a function of n, and
means that the quotient of two expressions tends to 1 as x approaches infinity.
Cn is C2 multiplied by a number which depends on the odd prime factors q of n: For example, C4 = C2 and C6 = 2C2.
It relies on some unproven assumptions so the conclusion remains a conjecture.
Now assume q divides n and consider a potential prime pair (a, a + n).
In the case of n = 6, the argument simplifies to: If a is a random number then 3 has a probability of 2/3 of dividing a or a + 2, but only a probability of 1/3 of dividing a and a + 6, so the latter pair is conjectured twice as likely to both be prime.