TheInfoList

A twin prime is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
that is either 2 less or 2 more than another prime numberâ€”for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a
prime gap A prime gap is the difference between two successive prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that i ...
of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime 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-called twin prime conjecture) or if there is a largest pair. The breakthrough work of
Yitang Zhang Yitang "Tom" Zhang () (born February 5, 1955) is a Chinese-born American mathematician working in the area of number theory. While working for the University of New Hampshire as a lecturer, Zhang submitted an article to the ''Annals of Mathematics ...

in 2013, as well as work by James Maynard, and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved.

Properties

Usually the pair (2, 3) is not considered to be a pair of twin primes. Since 2 is the only even 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 few twin prime pairs are: :(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), â€¦ . Five is the only prime that belongs to two pairs, as every twin prime pair greater than $\left(3, 5\right)$ is of the form $\left(6n-1, 6n+1\right)$ for some
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
''n''; that is, the number between the two primes is a multiple of 6. As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.

Brun's theorem

In 1915,
Viggo Brun Viggo Brun (13 October 1885 – 15 August 1978) was a Norway, Norwegian professor, mathematics, mathematician and Number theory, number theorist. Contributions In 1915, he introduced a new method, based on Adrien-Marie Legendre, Legendre's ve ...
showed that the sum of reciprocals of the twin primes was convergent. This famous result, called
Brun's theorem In number theory, Brun's theorem states that the sum of the Multiplicative inverse, reciprocals of the twin primes (pairs of prime numbers which differ by 2) Convergent series, converges to a finite value known as Brun's constant, usually denoted by ...
, was the first use of the
Brun sieveIn the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by Congruence relation#Modular arithmetic ...
and helped initiate the development of modern
sieve theory The fine mesh strainer, also known as the sift, commonly known as sieve, is a device for separating wanted elements from unwanted material or for characterizing the particle size distribution of a sample, typically using a woven screen such as ...
. The modern version of Brun's argument can be used to show that the number of twin primes less than ''N'' does not exceed :$\frac$ for some absolute constant ''C'' > 0.Bateman & Diamond (2004) p. 313 In fact, it is bounded above by :$\frac\left\left(1 + O\left\left(\frac\right\right)\right\right),$ where $C\text{'} = 8C_2$, where ''C''2 is the twin prime constant, given
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926â€“1988), American blues drummer *Fritz von Below (1853â ...
.

Twin prime conjecture

The question of whether there exist infinitely many twin primes has been one of the great open questions in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777â€“1855) said, "Mathematics is the queen of the sciencesâ€”and number theory is the queen ...

for many years. This is the content of the ''twin prime conjecture'', which states that there are infinitely many primes ''p'' such that ''p'' + 2 is also prime. In 1849,
de Polignac#Redirect DE {{Redirect category shell, 1= {{Redirect from other capitalisation {{Redirect from ambiguous term ...
made the more general conjecture that for every
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
''k'', there are infinitely many primes ''p'' such that ''p'' + 2''k'' is also prime. From p. 400: ''"1er ''ThÃ©orÃ¨me.'' Tout nombre pair est Ã©gal Ã  la diffÃ©rence de deux nombres premiers consÃ©cutifs d'une infinitÃ© de maniÃ¨res â€¦ "'' (1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways â€¦ ) The case ''k'' = 1 of
de Polignac's conjecture In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: :For any positive Parity (mathematics), even number ''n'', there are infinitely many prime gaps of size ''n''. In other words: There are infinitely many ca ...
is the twin prime conjecture. A stronger form of the twin prime conjecture, the Hardyâ€“Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the
prime number theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. On April 17, 2013,
Yitang Zhang Yitang "Tom" Zhang () (born February 5, 1955) is a Chinese-born American mathematician working in the area of number theory. While working for the University of New Hampshire as a lecturer, Zhang submitted an article to the ''Annals of Mathematics ...

announced a proof that for some integer ''N'' that is less than 70 million, there are infinitely many pairs of primes that differ by ''N''. Zhang's paper was accepted by ''
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal In academic publishing Academic publishing is the subfield of publishing Publishing is the activity of making information, literature, music, software and other content available ...
'' in early May 2013. subsequently proposed a
Polymath Project The Polymath Project is a collaboration among mathematicians to solve important and difficult Mathematics, mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The projec ...
collaborative effort to optimize Zhang's bound. As of April 14, 2014, one year after Zhang's announcement, the bound has been reduced to 246. Further, assuming the
Elliottâ€“Halberstam conjectureIn number theory, the Elliottâ€“Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who state ...
and its generalized form, the Polymath project wiki states that the bound has been reduced to 12 and 6, respectively. 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. A strengthening of
Goldbachâ€™s conjecture Goldbach's conjecture is one of the oldest and best-known list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even integer, whole number greater than 2 is ...
, if proved, would also prove there is an infinite number of twin primes.

Other theorems weaker than the twin prime conjecture

In 1940,
Paul ErdÅ‘s Paul ErdÅ‘s ( hu, ErdÅ‘s PÃ¡l ; 26 March 1913 â€“ 20 September 1996) was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. He was known both for his s ...

showed that there is a
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
''c'' < 1 and infinitely many primes ''p'' such that where ''p''â€² denotes the next prime after ''p''. What this means is that we can find infinitely many intervals that contain two primes 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 grow logarithmically. This result was successively improved; in 1986
Helmut Maier Helmut Maier (born 17 October 1953) is a Germany, German mathematician and professor at the University of Ulm, Germany. He is known for his contributions in analytic number theory and mathematical analysis and particularly for the so-called Maier's ...

showed that a constant ''c'' < 0.25 can be used. In 2004
Daniel Goldston Daniel Alan Goldston (born January 4, 1954 in Oakland, California Oakland is the largest city and the county seat A county seat is an administrative centerAn administrative centre is a seat of regional administration or local government ...

and
Cem YÄ±ldÄ±rÄ±m Cem YalÃ§Ä±n YÄ±ldÄ±rÄ±m (born 8 July 1961) is a Turkey, Turkish mathematician who specializes in number theory. He obtained his B.Sc from Middle East Technical University in Ankara, Turkey and his PhD from the University of Toronto in 1990. His a ...
showed that the constant could be improved further to ''c'' = 0.085786â€¦ In 2005, Goldston,
JÃ¡nos Pintz JÃ¡nos Pintz (born 20 December 1950 in Budapest) is a Hungary, Hungarian mathematician working in analytic number theory. He is a fellow of the AlfrÃ©d RÃ©nyi Institute of Mathematics, RÃ©nyi Mathematical Institute and is also a member of the Hung ...
and YÄ±ldÄ±rÄ±m established that ''c'' can be chosen to be arbitrarily small, i.e. :$\liminf_\frac=0.$ 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, . By assuming the
Elliottâ€“Halberstam conjectureIn number theory, the Elliottâ€“Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who state ...
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. The result of
Yitang Zhang Yitang "Tom" Zhang () (born February 5, 1955) is a Chinese-born American mathematician working in the area of number theory. While working for the University of New Hampshire as a lecturer, Zhang submitted an article to the ''Annals of Mathematics ...

, :
Polymath Project The Polymath Project is a collaboration among mathematicians to solve important and difficult Mathematics, mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution. The projec ...
optimization of Zhang's bound and the work of Maynard have reduced the bound to ''N'' = 246.

Conjectures

First Hardyâ€“Littlewood conjecture

The Hardyâ€“Littlewood conjecture (named after
G. H. Hardy Godfrey Harold Hardy (7 February 1877 â€“ 1 December 1947) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
and ) is a generalization of the twin prime conjecture. It is concerned with the distribution of
prime constellation In number theory, a prime ''k''-tuple is a finite collection of values representing a repeatable pattern of differences between prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), pr ...
s, including twin primes, in analogy to the
prime number theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. Let Ï€2(''x'') denote the number of primes ''p'' â‰¤ ''x'' such that ''p'' + 2 is also prime. Define the twin prime constant ''C''2 as :$C_2 = \prod_ \left\left(1 - \frac\right\right) \approx 0.66016 18158 46869 57392 78121 10014\dots$ (here the product extends over all prime numbers ''p'' â‰¥ 3). Then a special case of the first Hardy-Littlewood conjecture is that :$\pi_2\left(x\right) \sim 2 C_2 \frac \sim 2 C_2 \int_2^x$ in the sense that the quotient of the two expressions 1 as ''x'' approaches infinity.Bateman & Diamond (2004) pp.334â€“335 (The second ~ is not part of the conjecture and is proven by
integration by parts In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

.) The conjecture can be justified (but not proven) by assuming that 1 / ln ''t'' describes the
density function In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
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 Ï€2(''x'') above. The fully general first Hardyâ€“Littlewood conjecture on prime ''k''-tuples (not given here) implies that the is false. This conjecture has been extended by
Dickson's conjectureIn number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by that for a finite set of linear forms , , ..., with , there are infinitely many positive integers for which they are all prime number, prime, unless there ...
.

Polignac's conjecture

Polignac's conjecture In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: :For any positive even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer's parity is even if it is ...
from 1849 states that for every positive even natural number ''k'', there are infinitely many consecutive prime pairs ''p'' and ''pâ€²'' such that ''p''â€² âˆ’ ''p'' = ''k'' (i.e. there are infinitely many
prime gap A prime gap is the difference between two successive prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that i ...
s of size ''k''). The case ''k'' = 2 is the twin prime conjecture. 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 ''p''''n''+1 âˆ’ ''p''''n'' = ''m'' for all ''m'' < ''N'' and so for ''n'' large enough we have ''p''''n''+1 âˆ’ ''p''''n'' > ''N'', which would contradict Zhang's result.

Large twin primes

Beginning in 2007, two
distributed computing Distributed computing is a field of computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Comp ...
projects, Twin Prime Search and
PrimeGrid PrimeGrid is a volunteer distributed computing Distributed computing is a field of computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as pract ...
, have produced several record-largest twin primes. , the current largest twin prime pair known is 2996863034895 Â· 21290000 Â± 1, with 388,342 decimal digits. It was discovered in September 2016. There are 808,675,888,577,436 twin prime pairs below 1018. An empirical analysis of all prime pairs up to 4.35 Â· 1015 shows that if the number of such pairs less than x is f(x)Â·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 () (not to be confused with
Brun's constant In number theory, Brun's theorem states that the sum of the Multiplicative inverse, reciprocals of the twin primes (pairs of prime numbers which differ by 2) Convergent series, converges to a finite value known as Brun's constant, usually denoted by ...

), according to the Hardyâ€“Littlewood conjecture.

Other elementary properties

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3. Five is therefore the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a
Chen prime A prime number ''p'' is called a Chen prime if ''p'' + 2 is either a prime or a semiprime, product of two primes (also called a semiprime). The even number 2''p'' + 2 therefore satisfies Chen's theorem. The Chen primes are named after Ch ...
. It has been proven that the pair (''m'', ''m'' + 2) is a twin prime if and only if :$4\left(\left(m-1\right)! + 1\right) \equiv -m \pmod .$ If ''m'' âˆ’ 4 or ''m'' + 6 is also prime then the three primes are called a
prime tripletIn mathematics, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (''p'', ''p'' + 2, ''p'' + 6) or (''p'', ''p'' + 4, ' ...
. For a twin prime pair of the form (6''n'' âˆ’ 1, 6''n'' + 1) for some natural number ''n'' > 1, ''n'' must have units digit 0, 2, 3, 5, 7, or 8 ().

Isolated prime

An isolated prime (also known as single prime or non-twin prime) is a prime number ''p'' such that neither ''p'' âˆ’ 2 nor ''p'' + 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 both
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials * ...
. The first few isolated primes are :, 23, 37, 47, 53, 67, 79, 83, 89, 97, ... It follows from
Brun's theorem In number theory, Brun's theorem states that the sum of the Multiplicative inverse, reciprocals of the twin primes (pairs of prime numbers which differ by 2) Convergent series, converges to a finite value known as Brun's constant, usually denoted by ...
that
almost all In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no genera ...
primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold ''n'' and the number of all primes less than ''n'' tends to 1 as ''n'' tends to infinity.

*
Cousin primeIn mathematics, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. The cousin primes (sequences and in OEIS) b ...
*
Prime gap A prime gap is the difference between two successive prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that i ...
* Prime ''k''-tuple *
Prime quadrupletA prime quadruplet (sometimes called prime quadruple) is a set of four primes of the form . This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4. Prime quadruplets The first ...
*
Prime tripletIn mathematics, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form (''p'', ''p'' + 2, ''p'' + 6) or (''p'', ''p'' + 4, ' ...
*
Sexy prime Sexy primes are prime numbers A prime number (or a prime) is a natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the '' ...

References

*

*

*
Top-20 Twin Primes
at Chris Caldwell's
Prime Pages The PrimePages is a website about prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is call ...
* Xavier Gourdon, Pascal Sebah
''Introduction to Twin Primes and Brun's Constant''

"Official press release"
of 58711-digit twin prime record *
The 20 000 first twin primes

Polymath: Bounded gaps between primes

Sudden Progress on Prime Number Problem Has Mathematicians Buzzing
{{Prime number conjectures Classes of prime numbers Unsolved problems in number theory