Twin prime number

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Unresolved problems of mathematics: Are there infinite twin prime numbers?

Mathematical Landscape (grafed) of finite scale of prime-composed numbers using data obtained for x = 2 to 64. The lower chart symbolically represents "Dimensions" using increasingly large negative integers.

In mathematics, and more specifically in number theory, two prime numbers (p, q) are twin prime numbers if, being q > p, q – p = 2 holds. All prime numbers except 2 are odd. The only two consecutive prime numbers are 2 and 3. The question arises from finding two prime numbers that are consecutive odd, that is, the difference from the largest to the smallest is 2. The first to call them that way was Paul Stäckel.

Twin cousin duos

There are 35 pairs of twin prime numbers among the integers less than 1000, and they are (A077800):

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463), (521, 523), (569, 571), (599, 601), (617, 619), (641, 643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859), (881, 883).

Great Twin Primes

As of 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record-breaking twin prime numbers. As of August 2022, the largest known twin prime pair is 2996863034895 2^1290000 ± 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 10¹⁵ shows that if the number of such pairs less than x is f(x ) x/(log x)², so f(x) is approximately 1.7 for small x and decreases towards about 1.3 as x approaches infinity. The limiting value of f(x) is conjectured to be equal to twice the twin prime constant (A114907) (not to be confused with Brun's constant), by the Hardy-Littlewood conjecture.

Isolated cousin

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 number, since 21 and 25 are both composite.

Isolated first cousins are

2, 23, 37, 47, 53, 67, 79, 83, 89, 97,... A007510

From Brun's theorem it follows that almost all primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold n to the number of all primes less than n approaches 1 as n tends to infinity.

Properties

Typically, the pair (2, 3) is not considered 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; therefore, twin cousins are as close together as possible for any other pair of cousins.

Five is the only cousin who belongs to two pairs, since all pairs of twin cousins older than ${displaystyle (3,5)}$ is shape ${displaystyle (6n-1,6n+1)}$ for some natural number n; that is, the number between the two cousins is a multiple of 6. As a result, the sum of any pair of twin cousins (not 3 or 5) is divisible by 12.

In other words, from the pair (5, 7), the intermediate number is always a multiple of 6, therefore of 2 and 3.

Every third odd number is divisible by 3, which requires that three successive odd numbers cannot be prime unless one of them is 3. Therefore, five is the only prime that is part of two twin prime pairs. The lower member of a pair is, by definition, a Chen prime number.

The pair (m, m + 2) has been shown to be twin prime if and only if

{displaystyle 4((m-1)!+1)equiv -m{pmod {m(m+2)}}.}

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

For a twin prime pair of the form (6n − 1, 6n + 1) for some natural number n > 1, n must have the units digit 0, 2, 3, 5, 7, or 8 (A002822).

In 1915, Viggo Brun proved that the sum of the reciprocals of all twin prime numbers converges to a number, now called Brun's constant and denoted B₂:

{displaystyle B_{2}=left({frac {1}{3}}+{frac {1}{5}}right)+left({frac {1}{5}}+{frac {1}{7}}right)+left({frac {1}{11}}+{frac {1}{13}}right)+left({frac {1}{17}}+{frac {1}{19}}right)+left({frac {1}{29}}+{frac {1}{31}}right)+cdots }

This is in contrast to the sum of inverses of all primes, which diverges. Calculating the twin primes up to 10¹⁴ (and at the same time discovering the division error of the Intel Pentium), Thomas Nicely estimated Brun's constant at 1.902160578. The best estimate to date is that of Pascal Sebah and Patrick Demichel published in 2002, with all twin cousins up to 10¹⁶, obtaining 1.902160583104 as an approximation.

The pair n, n + 2 has been shown to be twin primes if and only if:

{displaystyle 4((n-1)!+1)equiv -n{pmod {n(n+2)}}}

Distribution of twin prime numbers

It is not known if there are infinitely many twin primes, although it is widely believed that they do. This is the content of the Twin Primes Conjecture. A strong form of the twin primes conjecture, the Hardy-Littlewood conjecture, postulates a distribution law for twin primes similar to the prime number theorem:

{displaystyle pi _{2}(x)sim 2C_{2}int _{2}^{x}{dt over (ln t)^{2}}}

where C₂ is the constant of twin primes, defined by the following Euler product:

{displaystyle prod _{textstyle {p;{rm {primo}} atop pgeq 3}}left(1-{frac {1}{(p-1)^{2}}}right)=0,66016118ldots }

The largest known twin primes are the pair 2996863034895 2^1290000 - 1 and 2996863034895 2^1290000 + 1, which have 388342 digits and were discovered in September 2016

There are 808,675,888,577,436 pairs of twin primes less than 10¹⁸.

Propositions relating to twin cousins

Theorems

Brun's Theorem

Main article: Brun Theorem

In 1915, Viggo Brun proved that the sum of reciprocals of twin primes converged. This famous result, known as Brun's theorem, was the first use of Brun's sieve and helped initiate the development of modern riddle theory. The revised version of Brun's argument can be used to show that the number of twin primes less than N does not exceed

{displaystyle {frac {CN}{(log N)^{2}}}}

for some absolute constant C > 0. In fact, it is bounded above by

{displaystyle {frac {C'N}{(log N)^{2}}}left(1+Oleft({frac {log log N}{log N}}right)right),}

where ${displaystyle C'=8C_{2}}$ Where C₂ It's the constant premium twingiven in the first conjecture of Hardy-Littlewood.

Other theorems

In 1940, Paul Erdős proved that there exists a constant c < 1 and infinitely many primes p such that (p′ − p) < (c ln p) where p′ denotes the next prime after p. What this means is that infinitely many intervals can be found containing two prime numbers (p,p′) as long as these intervals are allowed to grow slowly in size as you move toward larger and larger primes. Here, "grow slowly" means that the length of these intervals can grow logarithmically. This result was successively improved, and in 1986 Helmut Maier showed that there is a constant c < 0.25. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be further improved to c = 0.085786… In 2005 Goldston, János Pintz and Yıldırım established that c can be chosen for that is arbitrarily small,, that is,

{displaystyle liminf _{nto infty }{frac {p_{n+1}-p_{n}}{log p_{n}}}=0.}

On the other hand, this result does not rule out that there are not infinitely many intervals containing two prime numbers if only the intervals are allowed to grow in size, such as 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 ns 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 an infinite number of n, at least two of the n, n + 2, n + 4 and n + 6 are prime.

The result of Yitang Zhang,

<math alttext="{displaystyle liminf _{nto infty }(p_{n+1}-p_{n})lim infn→ → ∞ ∞ (pn+1− − pn).NwithN=7× × 107,{displaystyle liminf _{nto infty }(p_{n+1}-p_{n})}{n;{text{ with }};N=7times 10^{7},}<img alt="{displaystyle liminf _{nto infty }(p_{n+1}-p_{n})

is a major improvement on the Goldston-Graham-Pintz-Yıldırım result. The Polymath Project's optimization of Zhang's limit and Maynard's work have lowered the limit: the lower limit is at most 246.

Conjectures

Prime Numbers Conjecture

The question of whether there are 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 p such that p + 2 is also prime. In 1849, de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime numbers p such that p + 2k is also prime. The case k = 1 of the Polignac 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 similar to that of the prime number theorem.

On April 17, 2013, Yitang Zhang announced a proof that for some integer N less than 70 million, there are infinitely many pairs of prime numbers that differ by N >. Zhang's paper was accepted by the Annals of Mathematics in early May 2013. Terence Tao subsequently proposed a collaborative effort at the Polymath Project to optimize Zhang's limit. On April 14, 2014, one year after Zhang's announcement, the limit was lowered to 246. These improved limits were discovered using a different approach that was simpler than Zhang's and discovered independently by James Maynard and Terence Tao.. This second approach also provided bounds for the smallest f(m) needed to ensure that an infinite number of intervals of width f(m) contain at least m primes. Furthermore (see also the next section), assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that the limit is 12 and 6, respectively.

A strengthening of Goldbach's conjecture, if proven, would also prove that there are an infinite number of twin primes, just like the existence of Siegel's zero.

First Hardy-Littlewood conjecture

The Hardy-Littlewood conjecture (named after Godfrey Harold Hardy and John Littlewood) is a generalization of the twin prime conjecture. It deals with the distribution of k-tuples of prime numbers, including twin primes, in analogy with the prime number theorem. Let π₂(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C₂ as

{displaystyle C_{2}=prod _{textstyle {p;{rm {primo}} atop pgeq 3}}left(1-{frac {1}{(p-1)^{2}}}right)approx 0.660161815846869573927812110014dots }

(here the product extends over all prime numbers p ≥ 3). So a special case of the first Hardy-Littlewood conjecture is that

{displaystyle pi _{2}(x)sim 2C_{2}{frac {x}{(ln x)^{2}}}sim 2C_{2}int _{2}^{x}{dt over (ln t)^{2}}}

in the sense that the quotient of the two expressions approaches 1 as x approaches infinity. (The second ~ is not part of the conjecture and is proven by integration methods.)

The conjecture can be justified (but not proven) by assuming that (1 / ln t) describes the pdf of the prime distribution. This assumption, suggested by the Prime Number Theorem, implies the Twin Prime Conjecture, as shown in the formula for π₂(x) above.

The first fully general Hardy-Littlewood conjecture about prime K-tuples (not given here) implies that the second Hardy-Littlewood conjecture is false.

This conjecture has been extended by the Dickson conjecture.

Polignac's conjecture

Main article: Polygnac conjecture

Polignac's conjecture of 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 differences between consecutive primes of size k). The case k = 2 is the twin prime conjecture. The conjecture has yet to be proven or disproved 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 k did not exist, then for any positive even natural number N there are at most a finite number of n such that p_n+1 − p_n = m for all m < N and so for n large enough we have to p_n+1 − p_n > N, which would contradict Zhang's result.

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