# عددان أوليان توأم

عددان أوليان توأم Twin prime هو عدد أولي، إما أقل بإثنين أو أكثر بإثنين، من عدد أولي آخر -- فعلى سبيل المثال الزوج (41، 43). 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 or there is a largest pair. The 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.

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## تاريخ

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 p such that p + 2 is also prime. In 1849, دى پولينياك made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case k = 1 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.

## السمات

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), … .

Five is the only prime in two distinct pairs.

Every twin prime pair except $(3,5)$  is of the form $(6n-1,6n+1)$  for some natural number 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.

### مبرهنة برون

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

${\frac {CN}{(\log N)^{2}}}$

for some absolute constant C > 0. In fact, it is bounded above by :${\frac {C'N}{(\log N)^{2}}}\left(1+O\left({\frac {\log \log N}{\log N}}\right)\right)$ , where $C'=8C_{2}$ , where C2 is the twin prime constant, given below.

## مبرهنات أخرى أضعف من حدسية العددين الأولين التوأم

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. What this means is that we can find infinitely many intervals that contain two primes (p,p′) 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 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, i.e.

$\liminf _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}=0.$

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, 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 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.

نتيجة Yitang Zhang,

$\liminf _{n\to \infty }(p_{n+1}-p_{n})

is a major improvement on the Goldston–Graham–Pintz–Yıldırım result. The Polymath Project optimization of Zhang's bound and the work of Maynard has reduced the bound to N = 246.

## حدسيات

### حدسية هاردي-لتل‌وود الأولى

The 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 π2(x) denote the number of primes px such that p + 2 is also prime. Define the twin prime constant C2 as

$C_{2}=\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\dots$

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

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

in the sense that the quotient of the two expressions tends to 1 as x approaches infinity. (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 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem and would imply the twin prime conjecture, but remains unresolved.

The fully general first Hardy–Littlewood conjecture on prime k-tuples (not given here) implies that the حدسية هاردي-لتل‌وود الثانية is false.

### حدسية پولينياك

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 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 pn+1 − pn = m for all m < N and so for n large enough we have pn+1 − pn > N, which would contradict Zhang's result.

## الأوليات التوائم الكبيرة

Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid, have produced several record-largest twin primes. اعتبارا من سبتمبر 2018, the current largest twin prime pair known is 2996863034895 · 21290000 ± 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(xx/(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), according to the Hardy–Littlewood conjecture.

## صفات ابتدائية أخرى

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.

It has been proven that the pair (mm + 2) is a twin prime if and only if

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

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 units digit 0, 2, 3, 5, 7, or 8 ().

## الأعداد الأولية المنعزلة

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, ...

It follows from Brun's theorem that almost all primes are members of this sequence.

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## المراجع

1. ^ Terry Tao, Small and Large Gaps Between the Primes
2. ^ The First 100,000 Twin Primes
3. ^ Caldwell, Chris K. "Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?". The Prime Pages. The University of Tennessee at Martin. Retrieved 2018-09-27.
4. ^ Brun, V. (1915), "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" (in German), Archiv for Mathematik og Naturvidenskab 34 (8): 3–19, ISSN 0365-4524
5. ^ Bateman & Diamond (2004) p. 313
6. ^ Heini Halberstam, and Hans-Egon Richert, Sieve Methods, p. 117, Dover Publications, 2010
7. ^ 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, doi:10.3792/pjaa.82.61 .
8. ^ 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, doi:10.1090/S0002-9947-09-04788-6
9. ^ Maynard, James (2015), "Small gaps between primes", Annals of Mathematics, Second Series 181 (1): 383–413, doi:10.4007/annals.2015.181.1.7
10. ^ Polymath, D. H. J. (2014), "Variants of the Selberg sieve, and bounded intervals containing many primes", Research in the Mathematical Sciences 1: Art. 12, 83, doi:10.1186/s40687-014-0012-7
11. ^ قالب:Cite OEIS -- A page of number theoretical constants
12. ^ Bateman & Diamond (2004) pp.334–335
13. ^ Caldwell, Chris K. "The Prime Database: 2996863034895*2^1290000-1".
14. ^
15. ^ Tomás Oliveira e Silva (7 April 2008). "Tables of values of pi(x) and of pi2(x)". Aveiro University. Retrieved 7 January 2011.