Constant of Brown

In Mathematical, the constant of Brown of the prime numbers twins (or more simply constant of Brown ) is the sum of the series Inverse S of the Prime numbers twins, i.e. couples of distant prime numbers of 2.

This constant car its name of the mathematician Brown Viggo which showed in 1919 that this series is convergent.

Definition

That is to say

(p_n, q_n) _ {N \ in \ mathbb {NR}}

the continuation of the couples of prime numbers twins. The first terms of this continuation are

(3, 5)

(5, 7)

(11, 13)

, etc

That is to say $(S_n) _ {N \ in \ mathbb {NR}}$ the continuation of the sums partial of the opposite of the $n$ first terms of the preceding continuation: $S_n= \ sum_ {k=0} ^n \ left (\ frac {1} {p_k} + \ frac {1} {q_k} \ right)$ . The corresponding series converges towards the constant of Brown, noted $B_2$ :

$B_2 = \ left (\ frac {1} {3} + \ frac {1} {5} \ right) + \ left (\ frac {1} {5} + \ frac {1} {7} \ right) + \ left (\ frac {1} {11} + \ frac {1} {13} \ right) + \ left (\ frac {1} {17} + \ frac {1} {19} \ right) + \ left (\ frac {1} {29} + \ frac {1} {31} \ right) + \ cdots$ .

With the difference of the series of the opposite of all the prime numbers which, it, diverges, this series is convergent. A divergence of the series would have made it possible to prove to the Conjecture of the prime numbers twins; insofar as it is convergent, this conjecture is still not proven.

Estimate

A first estimate of the constant of Brown was carried out by Shanks and Wrench in 1974 using the first twins up to 2 million. R.P. Brent calculated in the 1976 all prime numbers twins until 10¹¹ and improved the result.

A better estimate of the constant of Brown was carried out by Thomas Nicely in 1994 by a discovery method by calculating the prime numbers twins until 10¹⁴ (for the anecdote, T. Nicely highlighted at this occasion the Bogue of the division of Pentium). It improved thereafter this approximation by using the twins until 1,6×10¹⁵ and updated this approximation with the passing of years. In September 2006, it gave the following estimate:

B ₂ = 1,90216 05825 38 ± 0.00000 00014 00.

The best estimate of the decimal writing of the constant of Brown was carried out in 2002 by Pascal Sebah and Patrick Demichel by using all the prime numbers twins until 10¹⁶:

B ₂ ≈ 1,90216 05831 04.

The continuation of the figures of the constant of Brown in decimal writing is referred in OEIS like the sequence.

Generalization

There exists also a constant of Brown for the quadruplets of prime numbers . A quadruplet of first is a couple made up of twins first, separated from a distance from 4 (the possible short distance) that is to say

(p, p+2, p+6, p+8)

. The first quadruplets of first are (5, 7,11,13), (11, 13,17,19), (101, 103,107,109). The constant of Brown for the quadruplets of first, noted B ₄, is the sum of the opposite of all the prime numbers of the quadruplets:

$B_4 = \ left (\ frac {1} {5} + \ frac {1} {7} + \ frac {1} {11} + \ frac {1} {13} \ right)$

+ \ left (\ frac {1} {11} + \ frac {1} {13} + \ frac {1} {17} + \ frac {1} {19} \ right) + \ left (\ frac {1} {101} + \ frac {1} {103} + \ frac {1} {107} + \ frac {1} {109} \ right) + \ cdots

with the value:

B ₄ = 0,87058 83800 ± 0,00000 00005.

See too

External bonds

'' One twins enumeration and Brun' S constant '' (Thomas Nicely)
'' Introduction to twin premiums and Brun' S constant computation '' (Pascal Sebah)

References

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