Twin prime

A twin prime is a pair of two prime numbers whose distance is 2. The smallest prime twins (3, 5), ( 5, 7) and (11, 13).

History

The term twin prime was first used by Paul Stäckel.

Definition

Twin primes is called two primes and whose difference is. The prime number is also called a twin prime to prime.

Properties

With the exception of n = 1, the last digit of n is 0, 2, 3, 5, 7 or 8, as in the other case, would be one of two numbers 6n -1 and 6n 1 divisible by 5, and thus is not prime.

With an integer n can be any odd number in the form 30n 1, 30n 3, 5 30n, 30n 7, ..., 30n 25, 30n 27, 30n 29 (the latter also known as 30n - 1) represent. Primes (excluding 3 and 5), but never by a 7 forms 30n 3, 5 30n, 30n 9, 15 30n, 30n 21, 30n and 30n 25 27 since numbers of the 7 forms always through 3 or 5 are divisible.

Therefore, each twin primes (except (3, 5) and ( 5, 7) ) with an integer n exactly one of the three forms

Or the latter representation, to illustrate the symmetry to ( 30n 11, 30n 13), alternatively written as ( 30n - 13 30n -11).

Others

The smallest pair of twin primes (3, 5).

The largest currently (as of December 2011) is known pair of twin primes

These are numbers with 200,700 digits. The new record figures have almost twice as many digits as the numbers of the previous record from 2009. The pair of numbers found by the DC PrimeGrid project.

Two twin primes with the distance of four, ie sequences of the form is called prime quadruplets.

Open question

The greater the numbers you look, the less prime numbers can be found there. Although an infinite number of primes, it is uncertain whether there are infinitely many twin primes. The twin prime conjecture states that there are infinitely many twin primes. She is one of the major open questions in number theory.

Yitang Zhang ( University of New Hampshire) in May 2013 proved that there are infinitely many prime pairs whose distance from each other is a maximum of 70,000,000. Based on this approach could now be reduced to only 272, the number of 70,000,000. A further reduce that number down to 2 would prove the prime twins conjecture though; Experts consider this with the approach of Zhang discovered but impossible. Sharper results when Zhang was able to achieve in the November 2013 James Maynard ( a post-doctoral researcher at the University of Montreal), who pushed the limit with an alternative method of proof to 600. He extended the results also to higher k-tuples of primes and found here the existence of infinitely many primes of clusters with upper bounds for the distance.

While the sum of the reciprocals of the primes is divergent ( Leonhard Euler ), Viggo Brun proved in 1919 that the sum of the reciprocals of the twin primes converges. From this one can not conclude that there is at last, nor that there are infinitely many twin primes. The limit of the sum is called Brun's constant and is according to the latest estimate of 2002, approximately 1.902160583104.

GH Hardy and JE Littlewood presented in 1923 a conjecture about the asymptotic density of twin primes on ( and that of other prime constellations ) known as first Hardy - Littlewood conjecture or as a special case the same for twin primes. Thereafter, the number of primes less than X is asymptotically twins by the formula

With the twin prime constant ( sequence A005597 in OEIS )

Given. Since the primes asymptotically have a density according to the prime number theorem, the conjecture is quite plausible, and also numerically can confirm well the asymptotic form. But it is like the twin prime conjecture unproven. Because of the assumption of Hardy and Littlewood follows the twin prime conjecture, it is called strong twin prime conjecture.

After Paul Erdős had shown in 1940 that a positive constant c < 1 exists such that for infinitely many pairs of consecutive primes, the inequality, efforts were made to find smaller and smaller values ​​of c. The mathematician Dan Goldston and Cem Yıldırım published in 2003 a proof, which they claimed to have proved that c can be chosen arbitrarily small, so there would be small gaps between two consecutive prime numbers again and again in the infinite sequence of primes. Andrew Granville took place in the same year an error in the 25-page proof. In February 2005, Goldston, Yildirim and Pintz could submit a correction. This was reviewed by the then fault finders and counted as correct. The new evidence presented promises some number theorist believes to be an important step towards a proof of the twin prime conjecture.

A generalization of the twin prime conjecture is the conjecture of Polignac (Alphonse de Polignac, 1849): for every even number n, there are infinitely many prime numbers with neighboring distance n The conjecture is open. About the density of prime intervals n, there are analogous to the case n = 2, a conjecture of Hardy and Littlewood.