Polignac's conjecture

In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states:^[1]

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.^[2]

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.^[7]

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 ${\displaystyle \pi _{n}(x)}$  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

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

where C_n is a function of n, and ${\displaystyle \sim }$  means that the quotient of two expressions tends to 1 as x approaches infinity.^[8]

C₂ is the twin prime constant

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

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:

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

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 ${\displaystyle {\tfrac {q-1}{q-2}}}$ . 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 ${\displaystyle {\tfrac {2}{q}}}$ , since q divides one 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 ${\displaystyle {\tfrac {1}{q}}}$ . 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 ${\displaystyle {\tfrac {q-1}{q}}}$  divided by ${\displaystyle {\tfrac {q-2}{q}}}$ . This equals ${\displaystyle {\tfrac {q-1}{q-2}}}$  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.

Notes

1. de Polignac, A. (1849). "Recherches nouvelles sur les nombres premiers" [New research on prime numbers]. Comptes rendus (in French). 29: 397–401. From p. 400: "1^er Théorème. Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières … " (1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways … )
2. Tattersall, J.J. (2005), Elementary number theory in nine chapters, Cambridge University Press, ISBN 978-0-521-85014-8, p. 112
3. Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761. Zbl 1290.11128. (subscription required)
4. Klarreich, Erica (19 May 2013). "Unheralded Mathematician Bridges the Prime Gap". Simons Science News. Retrieved 21 May 2013.
5. Augereau, Benjamin (15 January 2014). "An old mathematical puzzle soon to be unraveled?". Phys.org. Retrieved 10 February 2014.
6. "Bounded gaps between primes". Polymath. Retrieved 2014-03-27.
7. "Bounded gaps between primes". Polymath. Retrieved 2014-02-21.
8. Bateman, Paul T.; Diamond, Harold G. (2004), Analytic Number Theory, World Scientific, p. 313, ISBN 981-256-080-7, Zbl 1074.11001.

References

• Alphonse de Polignac, Recherches nouvelles sur les nombres premiers. Comptes Rendus des Séances de l'Académie des Sciences (1849)
• Weisstein, Eric W. "de Polignac's Conjecture". MathWorld.
• Weisstein, Eric W. "k-Tuple Conjecture". MathWorld.