|Are there infinitely many twin primes?|
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 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 pairs p and p′ such that p′ − p = 2k. The case k = 1 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.
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 N does not exceed
for some absolute constant C > 0.
In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that (p′ − p) < (c ln p) where p′ denotes the next prime after p. This result was successively improved; in 1986 Helmut Maier showed that a constant c < 0.25 can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786… In 2005, Goldston, János Pintz and Yıldırım established that c can be chosen to be arbitrarily small
In fact, by assuming the Elliott–Halberstam conjecture 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.
Every twin prime pair except (3, 5) is of the form (6n − 1, 6n + 1) for some natural number n, and with the exception of n = 1, n must end in 0, 2, 3, 5, 7, or 8.
It has been proven that the pair (m, m+2) is a twin prime if and only if
If m − 4 or m + 6 is also prime then the 3 primes are called a prime triplet.
Largest known twin prime pair
On January 15, 2007 two distributed computing projects, Twin Prime Search and PrimeGrid found the largest known twin primes, 2003663613 · 2195000 ± 1. The numbers have 58711 decimal digits. Their discoverer was Eric Vautier of France.
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 then f(x) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity.
There are 808,675,888,577,436 twin prime pairs below 1018.
The limiting value of f(x) is conjectured to equal twice the twin prime constant (not to be confused with Brun's constant)
The twin prime conjecture would give a better approximation, as with the prime counting function, by
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), … (sequence A077800 in OEIS).
The only even prime is 2; except for the pair (2, 3), twin primes are as closely spaced as possible for two primes.
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 pairs. Along the same lines, other than the first pair, the number centered between the twin primes must always be divisible by 6. The lower member of a pair is by definition a Chen prime.
First Hardy–Littlewood conjecture
The Hardy–Littlewood conjecture (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 π2(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C2 as
This conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem.
Polignac's conjecture from 1849 states that for every positive even natural number k, there are infinitely many consecutive prime pairs p and p′ such that p′ − p = k (i.e. there are infinitely many prime gaps of size k). The case k = 2 is the twin prime conjecture. The conjecture has not been proved or disproved for any value of k.
An isolated 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
- McKee, Maggie (14 May 2013). "First proof that infinitely many prime numbers come in pairs". Nature. ISSN 0028-0836.
- Brun, V. (1915), "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare", Arch. f. Math. og Naturv. (in German) 34 (8): 3–19, ISSN 0365-4524, JFM 45.0330.16
- Bateman & Diamond (2004) p.313
- Goldston, Daniel Alan; Motohashi, Yoichi; Pintz, János; Yıldırım, Cem Yalçın (2006), "Small gaps between primes exist", Japan Academy. Proceedings. Series A. Mathematical Sciences 82 (4): 61–65, arXiv:math.NT/0505300, MR 2222213 .
- Goldston, D. A.; Graham, S. W.; Pintz, J.; Yıldırım, C. Y. (2009), "Small gaps between primes or almost primes", Transactions of the American Mathematical Society 361 (10): 5285–5330, arXiv:math.NT/0506067, doi:10.1090/S0002-9947-09-04788-6, MR 2515812 .
- "News Archive". PrimeGrid. 6 August 2009. Retrieved 2009-08-07.
- "The Prime Database: 65516468355*2^333333-1". Prime Pages. 13 August 2009. Retrieved 2009-08-14.
- "News Archive". PrimeGrid. 25 December 2011. Retrieved 2011-12-25.
- "The Prime Database: 3756801695685 · 2666669 - 1". Prime Pages. 25 December 2011. Retrieved 2011-12-25.
- Tomás Oliveira e Silva (7 April 2008). "Tables of values of pi(x) and of pi2(x)". Aveiro University. Retrieved 7 January 2011.
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- Bateman & Diamond (2004) pp.334–335
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- Hazewinkel, Michiel, ed. (2001), "Twins", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- 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.
- Weisstein, Eric W., "Twin Primes", MathWorld.
- The 20 000 first twin primes