Atwin prime is aprime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (17, 19) or(41, 43). In other words, a twin prime is a prime that has aprime gap of two. Sometimes the termtwin prime is used for a pair of twin primes; an alternative name for this isprime twin orprime pair.
Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-calledtwin prime conjecture) or if there is a largest pair. The breakthrough[1]work ofYitang Zhang in 2013, as well as work byJames Maynard,Terence Tao and others, has made substantial progress towardsproving that there are infinitely many twin primes, but at present this remains unsolved.[2]
Usually the pair(2, 3) is not considered to be a pair of twin primes.[3]Since 2 is the onlyeven prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes.
The first several twin prime pairs are
Five is the only prime that belongs to two pairs, as every twin prime pair greater than (3, 5) is of the form for somenatural numbern; that is, the number between the two primes is a multiple of 6.[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 ofreciprocals of the twin primes wasconvergent.[5]This famous result, calledBrun's theorem, was the first use of theBrun sieve and helped initiate the development of modernsieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less thanN does not exceed
for some absolute constantC > 0.[6]In fact, it is bounded above bywhere is thetwin prime constant (slightly less than 2/3),given below.[7]
The question of whether there exist infinitely many twin primes has been one of the greatopen questions innumber theory for many years. This is the content of thetwin prime conjecture, which states that there are infinitely many primesp such thatp + 2 is also prime. In 1849,de Polignac made the more general conjecture that for every natural numberk, there are infinitely many primesp such thatp + 2k is also prime.[8]Thecasek = 1 ofde Polignac's conjecture is the twin prime conjecture.
A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to theprime number theorem.
On 17 April 2013,Yitang Zhang announced a proof that there exists anintegerN 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 aPolymath 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 byJames Maynard andTerence Tao. This second approach also gave bounds for the smallestf (m) needed to guarantee that infinitely many intervals of widthf (m) contain at leastm primes. Moreover (see also the next section) assuming theElliott–Halberstam conjecture and its generalized form, the Polymath Project wiki states that the bound is 12 and 6, respectively.[12]
A strengthening ofGoldbach’s conjecture, if proved, would also prove there is an infinite number of twin primes, as would the existence ofSiegel zeroes.
In 1940,Paul Erdős showed that there is aconstantc < 1 and infinitely many primesp such thatp′ −p <c lnp wherep′ denotes the next prime afterp. What this means is that we can find infinitely many intervals that contain two primes(p,p′) as long as we let these intervals grow slowly in size as we move to bigger and bigger primes. Here, "grow slowly" means that the length of these intervals can growlogarithmically. This result was successively improved; in 1986Helmut Maier showed that a constantc < 0.25 can be used. In 2004Daniel Goldston andCem Yıldırım showed that the constant could be improved further toc = 0.085786.... In 2005,Goldston,Pintz, andYıldırım established thatc 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 lnp.
By assuming theElliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely manyn such that at least two ofn,n + 2,n + 6,n + 8,n + 12,n + 18, orn + 20 are prime. Under a stronger hypothesis they showed that for infinitely manyn, at least two ofn,n + 2,n + 4, andn + 6 are prime.
The result ofYitang Zhang,
is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath Project optimization of Zhang's bound and the work of Maynard have reduced the bound: thelimit inferior is at most 246.[15][16]
Thefirst Hardy–Littlewood conjecture (named afterG. H. Hardy andJohn Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution ofprime constellations, including twin primes, in analogy to theprime number theorem. Let denote the number of primesp ≤x such thatp + 2 is also prime. Define thetwin prime constantC2 as[17](Here the product extends over all prime numbersp ≥ 3.) Then a special case of the first Hardy-Littlewood conjecture is thatin the sense that the quotient of the two expressionstends to 1 asx approaches infinity.[6] (The second ~ is not part of the conjecture and is proven byintegration by parts.)
The conjecture can be justified (but not proven) by assuming that describes thedensity 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 above.
The fully general first Hardy–Littlewood conjecture onprimek-tuples (not given here) implies that thesecond Hardy–Littlewood conjecture is false.
This conjecture has been extended byDickson's conjecture.
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Polignac's conjecture from 1849 states that for every positive even integerk, there are infinitely many consecutive prime pairsp andp′ such thatp′ −p =k (i.e. there are infinitely manyprime gaps of sizek). The casek = 2 is thetwin prime conjecture. The conjecture has not yet been proven or disproven for any specific value ofk, but Zhang's result proves that it is true for at least one (currently unknown) value ofk. Indeed, if such ak did not exist, then for any positive even natural numberN there are at most finitely manyn such that for allm <N and so forn large enough we have which would contradict Zhang's result.[8]
Beginning in 2007, twodistributed computing projects,Twin Prime Search andPrimeGrid, 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. It was discovered in September 2016.[19]
There are 808,675,888,577,436 twin prime pairs below 1018.[20][21]
An empirical analysis of all prime pairs up to 4.35 × 1015 shows that if the number of such pairs less thanx isf (x) ·x /(logx)2 thenf (x) is about 1.7 for smallx and decreases towards about 1.3 asx tends to infinity. The limiting value off (x) is conjectured to equal twice the twin prime constant (OEIS: A114907) (not to be confused withBrun's constant), according to the Hardy–Littlewood conjecture.
Every thirdodd number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition aChen prime.
Ifm − 4 orm + 6 is also prime then the three primes are called aprime triplet.
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 numbern > 1,n must end in the digit 0, 2, 3, 5, 7, or 8 (OEIS: A002822). Ifn were to end in 1 or 6, 6n would end in 6, and 6n −1 would be a multiple of 5. This is not prime unlessn = 1. Likewise, ifn were to end in 4 or 9, 6n would end in 4, and 6n +1 would be a multiple of 5. The same rule applies modulo any primep ≥ 5: Ifn ≡ ±6−1 (modp), then one of the pair will be divisible byp and will not be a twin prime pair unless 6n =p ±1.p = 5 just happens to produce particularly simple patterns in base 10.
Anisolated prime (also known assingle prime ornon-twin prime) is a prime numberp such that neitherp − 2 norp + 2 is prime. In other words,p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are bothcomposite.
The first few isolated primes are
It follows fromBrun's theorem thatalmost all primes are isolated in the sense that the ratio of the number of isolated primes less than a given thresholdn and the number of all primes less thann tends to 1 asn tends to infinity.
[From p. 400]"1erThéorème. 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 ...)
