Prime numbers twins

In Mathematical, two prime numbers twins are two prime numbers who differ only from two. Except the pair (2, 3), it is about the smallest possible distance between two prime numbers.

Among the prime numbers twins, one finds 5 and 7, 11 and 13 or 821 and 823.

Definition

That is to say

(p, Q)

a couple of integers such as

p

and

q

is both of the prime numbers and

p < q

. It is said that

(p, Q)

form a couple of prime numbers twins if

q = p + 2

Properties

By omitting the couple

(2, 3)

, 2 is the smallest possible distance between two prime numbers; two prime numbers twins are thus two odd numbers consecutive.

Any couple of prime numbers twins (except for the couple $(3,5)$ ) is form $(6n - 1,6n + 1)$ for a certain entirety $n$ . Any series of three consecutive natural integers comprises at least a multiple of 2 (possibly two) and only one multiple of 3; these two multiples are confused between the two prime numbers twins.

It is possible to show that, for entire $m \ Ge 2$ , the couple $(m, m + 2)$ consists of prime numbers twins if and only if $4 + 1 + m \ equiv 0 \ MOD m (m+2)$ . This modular characterization and Factorielle of the prime numbers twins was discovered by P.A. Clement in 1949.

The series of the Inverse S of prime numbers twins is convergent towards the Constante of Brown, contrary to the series of the opposite of prime numbers.

Conjecture prime numbers twins

The Conjecture of the prime numbers twins affirms that there exists an infinity of prime numbers twins. Although the majority of the researchers in Théorie of the numbers think that this conjecture is true, it shown forever. There exists also a stronger version of this conjecture: the conjecture of Hardy - Littlewood, which provides to a law of distribution of the prime numbers twins and who takes as a starting point the Théorème of the prime numbers.

First first twins

The whole of the prime numbers twins up to 1000:

Larger first twins

January 15th, 2007, two projects of distributed calculation, Twin Precedes Search and PrimeGrid , discovered the largest couple of prime numbers twins currently known (i.e. in January 2007). The discoverer is French Eric Vautier.

The couple record is 2003663613 × 2¹⁹⁵⁰⁰⁰±1; the two numbers have 58.711 digits in decimal writing.

See too

External bonds

'' Twin Premiums '' (Chris Caldwell)
'' Introduction to Twin Premiums and Brun' S Constant '' (Xavier Gourdon, Pascal Sebah)
Site of Primegrid. Project of Calculation distributed using BOINC in order to seek prime numbers twins.

References

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