The breakthrough[1] work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved.
[4] As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent.
[5] This famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory.
The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed for some absolute constant C > 0.
[7] The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years.
On 17 April 2013, Yitang Zhang announced a proof that there exists an integer N that is less than 70 million, where there are infinitely many pairs of primes that differ by N.[9] Zhang's paper was accepted in early May 2013.
[10] Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound.
[11] One year after Zhang's announcement, the bound had been reduced to 246, where it remains.
[12] These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by James Maynard and Terence Tao.
This second approach also gave bounds for the smallest f (m) needed to guarantee that infinitely many intervals of width f (m) contain at least m primes.
Moreover (see also the next section) assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath Project wiki states that the bound is 12 and 6, respectively.
[12] A strengthening of Goldbach’s conjecture, if proved, would also prove there is an infinite number of twin primes, as would the existence of Siegel zeroes.
This result was successively improved; in 1986 Helmut Maier showed that a constant c < 0.25 can be used.
In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786... .
In 2005, Goldston, Pintz, and Yıldırım established that c can be chosen to be arbitrarily small,[13][14] i.e. On the other hand, this result does not rule out that there may not be infinitely many intervals that contain two primes if we only allow the intervals to grow in size as, for example, c ln ln p .
By assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many n such that at least two of n, n + 2, n + 6, n + 8, n + 12, n + 18, or n + 20 are prime.
Under a stronger hypothesis they showed that for infinitely many n, at least two of n, n + 2, n + 4, and n + 6 are prime.
in the sense that the quotient of the two expressions tends to 1 as x approaches infinity.
describes the density function of the prime distribution.
This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for
Polignac's conjecture from 1849 states that for every positive even integer k, there are infinitely many consecutive prime pairs p and p′ such that p′ − p = k (i.e. there are infinitely many prime gaps of size k).
The conjecture has not yet been proven or disproven for any specific value of k, but Zhang's result proves that it is true for at least one (currently unknown) value of k. Indeed, if such a k did not exist, then for any positive even natural number N there are at most finitely many n such that
[8] Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record-largest twin primes.
As of January 2025[update], the current largest twin prime pair known is 2996863034895 × 21290000 ± 1 ,[18] with 388,342 decimal digits.
[20][21] An empirical analysis of all prime pairs up to 4.35 × 1015 shows that if the number of such pairs less than x is f (x) ·x /(log x)2 then f (x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity.
The limiting value of f (x) is conjectured to equal twice the twin prime constant (OEIS: A114907) (not to be confused with Brun's constant), according to the Hardy–Littlewood conjecture.
The lower member of a pair is by definition a Chen prime.
It has been proven[22] that the pair (m, m + 2) is a twin prime if and only if For a twin prime pair of the form (6n − 1, 6n + 1) for some natural number n > 1, n must end in the digit 0, 2, 3, 5, 7, or 8 (OEIS: A002822).
In other words, p is not part of a twin prime pair.