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*.^{ }

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.^{
}

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; the Bateman–Horn conjecture gives conjectured asymptotic densities.

Let 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 means that the quotient of two expressions tends to 1 as *x* approaches infinity.^{[citation needed]}

*C*_{2} is the twin prime constant

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

*C _{n}* is

For example, *C*_{4} = *C*_{2} and *C*_{6} = 2*C*_{2}. 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 . 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 , 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 . 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 divided by . This equals 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.