Twin prime

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 or 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

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.

Properties

Usually the pair 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 several twin prime pairs are

: .

Five is the only prime that belongs to two pairs, as every twin prime pair greater than is of the form <math>(6n-1, 6n+1)</math> for some natural number ; that is, the number that falls 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 showed that the sum of reciprocals of the twin primes was convergent.

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 does not exceed

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

for some absolute constant

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.

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 such that is also prime. In 1849, de Polignac made the more general conjecture that for every natural number , there are infinitely many primes such that is also prime.

The of de Polignac's conjecture is the twin prime conjecture.

A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, 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 that is less than 70 million, where there are infinitely many pairs of primes that differ by . Zhang's paper was accepted in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to improve Zhang's bound.

One year after Zhang's announcement, the bound had been reduced to 246, where it remains.

These improved bounds were discovered independently by James Maynard and Terence Tao, using a different approach that was simpler than Zhang's. This second approach also gave bounds for the smallest needed to guarantee that infinitely many intervals of width contain at least 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.

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,

By assuming the Elliott–Halberstam conjecture or a slightly weaker version, they were able to show that there are infinitely many such that at least two of , , , , , , or are prime. Under a stronger hypothesis they showed that for infinitely many , at least two of , , , and 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 improvement of Zhang's bound and the work of Maynard have reduced the bound: the limit inferior is at most 246.

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 denote the number of primes such that is also prime. Define the twin prime constant as

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 .) Then a special case of the first Hardy–Littlewood conjecture is that

\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 approaches infinity. with 388,342 decimal digits. It was discovered in September 2016.

There are 808,675,888,577,436 twin prime pairs below .

An empirical analysis of all prime pairs up to 4.35 × shows that if the number of such pairs less than is then is about 1.7 for small and decreases towards about 1.3 as tends to infinity. The limiting value of is conjectured to equal twice the twin prime constant () (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 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 (). 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<sup>−1</sup> (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.

<span class="anchor" id="Isolated prime"></span>Isolated prime

An isolated prime (also known as a 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, ... .

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.

References

External links

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