Twin prime

Template:Short description A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair Template:Nobr or Template:Nobr In other words, a twin prime is a prime that has a prime gap 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<ref> Template:Cite periodical </ref> 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.<ref> Template:Cite AV media </ref> Template:Unsolved

Properties

Usually the pair Template:Math is not considered to be a pair of twin primes.<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref> 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 several twin prime pairs are

Template:Math Template:Math Template:Math Template:Math Template:OEIS2C.

Five is the only prime that belongs to two pairs, as every twin prime pair greater than Template:Math is of the form <math>(6n-1, 6n+1)</math> for some natural number Template:Mvar; that is, the number between the two primes is a multiple of 6.<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref> As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.

Brun's theorem

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent.<ref name=Brun-1915> Template:Cite journal </ref> 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 Template:Mvar does not exceed

<math>\frac{CN}{(\log N)^2}</math>

for some absolute constant Template:Nobr<ref name="Bateman-Diamond-2004"> Template:Cite book </ref> In fact, it is bounded above by <math display=block>\frac{8 C_2 N}{(\log N)^2} \left[ 1 + \operatorname{\mathcal O}\left(\frac{\log \log N}{\log N} \right) \right],</math> where <math>C_2</math> is the twin prime constant (slightly less than 2/3), given below.<ref> Template:Cite book </ref>

Twin prime conjecture

The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states that there are infinitely many primes Template:Mvar such that Template:Nobr is also prime. In 1849, de Polignac made the more general conjecture that for every natural number Template:Mvar, there are infinitely many primes Template:Mvar such that Template:Nobr is also prime.<ref name=dePolignac-1849> Template:Cite journal </ref> The Template:Nobr of de 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 the prime number theorem.

On 17 April 2013, Yitang Zhang announced a proof that there exists an integer Template:Mvar that is less than 70 million, where there are infinitely many pairs of primes that differ by Template:Mvar.<ref> Template:Cite news </ref> Zhang's paper was accepted in early May 2013.<ref> Template:Cite journal </ref> Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound.<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref>

One year after Zhang's announcement, the bound had been reduced to 246, where it remains.<ref name=nielsen-bd-gaps> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref> 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 Template:Nobr needed to guarantee that infinitely many intervals of width Template:Math contain at least Template:Mvar 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.<ref name=nielsen-bd-gaps/>

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.

Other theorems weaker than the twin prime conjecture

In 1940, Paul Erdős showed that there is a constant Template:Math and infinitely many primes Template:Mvar such that Template:Math where Template:Mvar denotes the next prime after Template:Mvar. What this means is that we can find infinitely many intervals that contain two primes Template:Math 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 showed that a constant Template:Math can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to Template:Nobr In 2005, Goldston, Pintz, and Yıldırım established that Template:Mvar can be chosen to be arbitrarily small,<ref> Template:Cite journal </ref><ref> Template:Cite journal </ref> i.e.

<math>\liminf_{n\to\infty} \left( \frac{ p_{n+1} - p_n }{\log p_n} \right) = 0 ~.</math>

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, Template:Nobr

By assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many Template:Mvar such that at least two of Template:Mvar, Template:Math, Template:Math, Template:Math, Template:Math, Template:Math, or Template:Math are prime. Under a stronger hypothesis they showed that for infinitely many Template:Mvar, at least two of Template:Mvar, Template:Math, Template:Math, and Template:Math are prime.

The result of Yitang Zhang,

<math> \liminf_{n\to\infty} (p_{n+1} - p_n) < N ~ \mathrm{ with } ~ N=7 \times 10^7,</math>

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: the limit inferior is at most 246.<ref> Template:Cite journal </ref><ref> Template:Cite journal </ref>

Conjectures

First Hardy–Littlewood conjecture

The first Hardy–Littlewood conjecture (named after G. H. Hardy and John Littlewood) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations, including twin primes, in analogy to the prime number theorem. Let Template:Tmath denote the number of primes Template:Math such that Template:Math is also prime. Define the twin prime constant Template:Math as<ref>Template:Cite OEIS</ref> <math display="block"> C_2 = \prod_{\textstyle{p \; \mathrm{prime,}\atop p \ge 3}} \left(1 - \frac{1}{(p-1)^2} \right) \approx 0.66016 18158 46869 57392 78121 10014 \ldots . </math> (Here the product extends over all prime numbers Template:Math.) Then a special case of the first Hardy-Littlewood conjecture is that <math display="block"> \pi_2(x) \sim 2 C_2 \frac{x}{(\ln x)^2} \sim 2 C_2 \int_2^x {\mathrm{d} t \over (\ln t)^2} </math> in the sense that the quotient of the two expressions tends to 1 as Template:Mvar approaches infinity.<ref name=Bateman-Diamond-2004/> (The second ~ is not part of the conjecture and is proven by integration by parts.)

The conjecture can be justified (but not proven) by assuming that Template:Tmath 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 Template:Tmath above.

The fully general first Hardy–Littlewood conjecture on [[prime k-tuple|prime Template:Mvar-tuple]]s (not given here) implies that the second Hardy–Littlewood conjecture is false.

This conjecture has been extended by Dickson's conjecture.

Polignac's conjecture

Template:More citations needed Polignac's conjecture from 1849 states that for every positive even integer Template:Mvar, there are infinitely many consecutive prime pairs Template:Mvar and Template:Mvar such that Template:Math (i.e. there are infinitely many prime gaps of size Template:Mvar). The case Template:Math is the twin prime conjecture. The conjecture has not yet been proven or disproven for any specific value of Template:Mvar, but Zhang's result proves that it is true for at least one (currently unknown) value of Template:Mvar. Indeed, if such a Template:Mvar did not exist, then for any positive even natural number Template:Mvar there are at most finitely many Template:Mvar such that <math>p_{n+1} - p_n = m</math> for all Template:Math and so for Template:Mvar large enough we have <math>p_{n+1} - p_n > N,</math> which would contradict Zhang's result.<ref name=dePolignac-1849/>

Large twin primes

Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record-largest twin primes. Template:As of, the current largest twin prime pair known is Template:Nobr<ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref> with 388,342 decimal digits. It was discovered in September 2016.<ref> Template:Cite news </ref>

There are 808,675,888,577,436 twin prime pairs below Template:10^.<ref> Template:Cite OEIS </ref><ref> {{#invoke:citation/CS1|citation |CitationClass=web }} </ref>

An empirical analysis of all prime pairs up to 4.35 × Template:10^ shows that if the number of such pairs less than Template:Mvar is Template:Math then Template:Math is about 1.7 for small Template:Mvar and decreases towards about 1.3 as Template:Mvar tends to infinity. The limiting value of Template:Math is conjectured to equal twice the twin prime constant (Template:OEIS2C) (not to be confused with Brun's constant), according to the Hardy–Littlewood conjecture.

Other elementary properties

Every third odd 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 a Chen prime.

If m − 4 or m + 6 is also prime then the three primes are called a prime triplet.

It has been proven<ref>Template:Cite journal</ref> that the pair (m, m + 2) is a twin prime if and only if

<math>4((m-1)! + 1) \equiv -m \pmod {m(m+2)}.</math>

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 (Template:OEIS2C). If n 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 unless n = 1. Likewise, if n 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 prime p ≥ 5: If n ≡ ±6⁻¹ (mod p), then one of the pair will be divisible by p and will not be a twin prime pair unless 6n = p ±1. p = 5 just happens to produce particularly simple patterns in base 10.

Isolated prime

An isolated prime (also known as single prime or non-twin prime) is a prime number p such that neither p − 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.

The first few isolated primes are

2, 23, 37, 47, 53, 67, 79, 83, 89, 97, ... Template:OEIS2C.

It follows from Brun's theorem that almost all 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.

See also

• Cousin prime
• Prime gap
• Prime k-tuple
• Prime quadruplet
• Prime triplet
• Sexy prime

Further reading

• Template:Cite book

External links

• Template:Springer
• Top-20 Twin Primes at Chris Caldwell's Prime Pages
• Xavier Gourdon, Pascal Sebah: Introduction to Twin Primes and Brun's Constant
• "Official press release" of 58711-digit twin prime record
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:TwinPrimes%7CTwinPrimes.html}} |title = Twin Primes |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• The 20 000 first twin primes
• Polymath: Bounded gaps between primes
• Sudden Progress on Prime Number Problem Has Mathematicians Buzzing