Prime twin

Number of prime twin pairs less than or equal to n

A prime twin is a pair of prime numbers that are spaced 2 apart. The smallest prime twins are (3, 5), (5, 7), and (11, 13).

history

The term prime number twin was first used by Paul Stäckel .

definition

Each pair of two prime numbers and with the difference is called a prime number twin . ${\displaystyle (p_{1},\,p_{2})}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{2}}$ ${\displaystyle p_{2}-p_{1}=2}$

properties

Graphical representation:
Y-axis = n
X-axis = divisor of 6n-1 OR 6n + 1
point = 6n-1 OR 6n + 1 are divisible
Parallel = prime number twin

With the exception of the prime number twin, there is always a number divisible by 6 between the two prime numbers of a prime number twin. ${\displaystyle (3,5)}$

Any integer may be namely in the form of , , , , or represent, with an integer. Numbers of the form , and are divisible by 2 and therefore cannot be prime numbers with the exception of two. Numbers of the form or are divisible by 3 and therefore cannot be prime numbers with the exception of three. Thus, all prime numbers greater than 3 have the form or . It follows that every prime twin except for has the representation . ${\displaystyle 6n-2}$ ${\displaystyle 6n-1}$ ${\displaystyle 6n}$ ${\displaystyle 6n+1}$ ${\displaystyle 6n+2}$ ${\displaystyle 6n+3}$ ${\displaystyle n}$ ${\displaystyle 6n-2}$ ${\displaystyle 6n}$ ${\displaystyle 6n+2}$ ${\displaystyle 6n+3}$ ${\displaystyle 6n}$ ${\displaystyle 6n-1}$ ${\displaystyle 6n+1}$ ${\displaystyle (3,5)}$ ${\displaystyle (6n-1,6n+1)}$

n (6n-1) (6n + 1) 1 5 7th 2 11 13 3 17th 19th 5 29 31 7th 41 43 10 59 61 12 71 73 17th 101 103 18th 107 109 23 137 139 25th 149 151 30th 179 181

n (6n-1) (6n + 1) 32 191 193 33 197 199 38 227 229 40 239 241 45 269 271 47 281 283 52 311 313 58 347 349 70 419 421 72 431 433 77 461 463 87 521 523

n (6n-1) (6n + 1) 95 569 571 100 599 601 103 617 619 107 641 643 110 659 661 135 809 811 137 821 823 138 827 829 143 857 859 147 881 883 170 1019 1021 172 1031 1033

n (6n-1) (6n + 1) 175 1049 1051 177 1061 1063 182 1091 1093 192 1151 1153 205 1229 1231 213 1277 1279 215 1289 1291 217 1301 1303 220 1319 1321 238 1427 1429 242 1451 1453 247 1481 1483

n (6n-1) (6n + 1) 248 1487 1489 268 1607 1609 270 1619 1621 278 1667 1669 283 1697 1699 287 1721 1723 298 1787 1789 312 1871 1873 313 1877 1879 322 1931 1933 325 1949 1951 333 1997 1999

(Follow A001097 in OEIS ) and (Follow A077800 in OEIS )

With the exception of n = 1, the last digit of an n is a 0, 2, 3, 5, 7 or an 8, since in the other case one of the two numbers 6 n -1 or 6 n +1 can be divided by 5 and therefore none Would be prime.

With an integer n , every odd number can be expressed in the form 30 n +1, 30 n +3, 30 n +5, 30 n +7, ..., 30 n +25, 30 n +27, 30 n +29 ( represent the latter as 30 n -1). Prime numbers (except 3 and 5) are never of one of the 7 forms 30 n +3, 30 n +5, 30 n +9, 30 n +15, 30 n +21, 30 n +25 and 30 n +27, since numbers of these 7 forms are always divisible by 3 or by 5.

Therefore, every pair of twin prime numbers (except (3, 5) and (5, 7)) with an integer n has exactly one of the three forms

(30 n -1, 30 n +1), (30 n +11, 30 n +13), (30 n +17, 30 n +19)

or the latter representation, to clarify the symmetry of (30 n +11, 30 n +13), alternatively written as (30 n -13, 30 n -11).

Others

The smallest pair of prime twins is (3, 5); the prime numbers 2 and 3 with the distance 1 are by definition not a pair of prime twins.

The largest currently (as of September 19, 2016) known pair of prime twins is

2,996,863,034,895 · 2 ^1,290,000 ± 1

these are numbers with 388,342 digits. The new record numbers have almost twice as many digits as the previous record from 2011. The pair of numbers was found by the PrimeGrid volunteer computing project .

Two prime twins with a distance of four, i.e. sequences of the form are called prime quadruplets . ${\displaystyle (p,p+2,p+6,p+8),}$

Open question

The larger numbers you look at, the fewer prime numbers you will find there. Although there are infinitely many prime numbers, it is uncertain whether there are infinitely many prime twins. The twin prime conjecture states that there are infinitely many twins prime. It is one of the great open questions in number theory.

While the sum of the reciprocal values ​​of the prime numbers is divergent ( Leonhard Euler ), Viggo Brun proved in 1919 that the sum of the reciprocal values ​​of the prime twins converges. From this one can neither conclude that there are finite nor that there are infinitely many prime twins. The limit of the sum is called the Brunsche constant and, according to the latest estimate from 2002, is around 1.902160583104.

GH Hardy and JE Littlewood put forward a conjecture about the asymptotic density of prime twins (and that of other prime number constellations) in 1923 , known as the first Hardy-Littlewood conjecture or as a special case of the same for prime twins. After that, the number of prime twins is less than asymptotically given by the formula ${\displaystyle x}$

${\displaystyle 2\,C_{2}\int _{2}^{x}\!{\frac {\mathrm {d} t}{(\log t)^{2}}}}$

with the prime number twin constant (sequence A005597 in OEIS )

${\displaystyle C_{2}=\prod _{p>2 \atop p\;{\text{prim}}}\left(1-{\frac {1}{(p-1)^{2}}}\right)=0{,}66016\ 18158\ 46869\ 57392\dots }$

given. Since the prime numbers have a density asymptotically according to the prime number theorem , the assumption is entirely plausible, and the asymptotic form can also be confirmed numerically. But like the twin prime conjecture, it is unproven. Since Hardy and Littlewood's conjecture gives rise to the twin prime conjecture, it is also called the strong twin prime conjecture . ${\displaystyle {\tfrac {1}{\log t}}}$

After Paul Erdős had shown in 1940 that a positive constant exists, so that the inequality holds for an infinite number of pairs of consecutive prime numbers , one tried to find ever smaller values ​​for . The mathematicians Dan Goldston and Cem Yıldırım published a proof in 2003 with which they claimed to have proven that one can choose arbitrarily small, which means that in the infinite sequence of prime numbers there would always be small distances between two consecutive prime numbers. Andrew Granville found a mistake in the 25-page proof that same year. In February 2005 Goldston, Yıldırım and Pintz were able to submit a correction. This was checked by the bug finders at the time and rated as correct. According to some number theorists, the newly presented proof promises to be an important step towards a proof of the twin prime conjecture. ${\displaystyle c<1}$ ${\displaystyle p}$ ${\displaystyle p'}$ ${\displaystyle p'-p<c\log(p)}$ ${\displaystyle c}$ ${\displaystyle c}$

A generalization of the twin prime conjecture is Polignac's conjecture ( Alphonse de Polignac , 1849): for every even number there are infinitely many neighboring prime numbers with spacing . The assumption is open. Analogous to the case, there is a conjecture by Hardy and Littlewood about the density of the distance between prime numbers . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n=2}$

Yitang Zhang (University of New Hampshire) proved in May 2013 that there are infinitely many pairs of prime numbers, the distance between which is a maximum of 70,000,000. Based on this approach, the number has now been reduced from 70,000,000 to just 246. Reducing this number further down to 2 would prove the twin prime conjecture; Experts consider this impossible with the approach discovered by Zhang. In November 2013, James Maynard (a post-doctoral student at the University of Montreal) achieved sharper results than Zhang , who pushed the limit to 600 with an alternative method of proof. He extended the results to higher tuples of prime numbers and also found here the existence of an infinite number of clusters of prime numbers with upper bounds for the distance. ${\displaystyle k}$

There are also related questions in function bodies.

Isolated prime number

An isolated prime number (from isolated prime , single prime or non-twin prime ) is a prime number for which the following applies: ${\displaystyle p\in \mathbb {P} }$

Neither nor is it a prime number. ${\displaystyle p-2}$ ${\displaystyle p+2}$

In other words: is not part of a prime twin. ${\displaystyle p}$

Examples

• The number is an isolated prime because and are not prime. ${\displaystyle p=23}$ ${\displaystyle p-2=21=3\cdot 7\not \in \mathbb {P} }$ ${\displaystyle p+2=25=5\cdot 5\not \in \mathbb {P} }$
• The smallest isolated prime numbers are the following:

2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, ... (sequence A007510 in OEIS )

properties

• Almost all prime numbers are isolated prime numbers. ( almost all of them are meant in a number-theoretic sense)
• There are infinitely many isolated prime numbers (follows from the above property).

Generalizations

A generalization of prime twins are prime tuples.

literature

• Wolfgang Blum: Goldbach and the twins . In: Spektrum der Wissenschaft , Dossier 6/2009: “The greatest riddles in mathematics”, ISBN 978-3-941205-34-5 , pp. 34–39.

Web links

Commons : Twin primes  - collection of images, videos and audio files

• Eric W. Weisstein : Twin Primes . In: MathWorld (English).
• Jeffrey F. Gold, Don H. Tucker: A characterization of twin prime pairs . (PDF; 123 kB) In: Proc. Fifth Nat. Conf. Undergrad. Res. , 1991, Volume I, pp. 362-366 (English)
• The Top Twenty: Twin Primes . - The 20 largest known prime twins
• Video: Prime Twins . Heidelberg University of Education (PHHD) 2012, made available by the Technical Information Library (TIB), doi : 10.5446 / 19865 .

Individual evidence

1. ↑ Twin Prime Records
2. ^ GH Hardy , JE Littlewood : Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes . (PDF; 2.5 MB) In: Acta Mathematica , 44, 1923, pp. 1–70 (English)
3. ↑ Eric W. Weisstein : Twin Prime Conjecture . In: MathWorld (English).
4. ^ Paul Erdős : The difference of consecutive primes . In: Duke Mathematical Journal , 6, 1940, pp. 438-441 (English). See Jerry Li: Erdos and the twin prime conjecture . (PDF; 157 kB) June 2, 2010 (English)
5. ↑ DA Goldston , J. Pintz , CY Yıldırım : Primes in tuples I . arxiv : math.NT / 0508185 , 2005 (English); simplified in DA Goldston, Y. Motohashi, J. Pintz, CY Yıldırım: Small gaps between primes exist . In: Proceedings of the Japan Academy , Series A 82, 2006, pp. 61–65 (English)
6. ↑ May 2005: Breakthrough in Prime Number theory at the American Institute of Mathematics (English)
7. ^ Polignac Conjecture . Mathworld
8. ^ Nature Online, 2013
9. ↑ Mathematics: Chinese succeeds in proving prime twins . Spiegel Online , May 22, 2013
10. ↑ News from number theory: A proof of the prime number twin conjecture is getting closer - Science Backgrounds. Neue Zürcher Zeitung , May 22, 2013
11. ↑ Bounded gaps between primes
12. ^ Terence Tao Bounded gaps between primes (Polymath8) - a progress report .
13. ↑ James Maynard: Small gaps between primes . arxiv : 1311.4600 preprint 2013
14. ↑ Erica Klarreich: Together and alone, solving the prime gap ( Memento of the original from November 20, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Simons Foundation, 2013 
15. ↑ Lior Bary-Soroker, Prime tuples in function fields , Mathematical Snapshots, Oberwolfach 2016
16. ^ Neil Sloane : Single (or isolated or non-twin) primes - Comments. OEIS , accessed August 2, 2018 .