Brun's constant

The Brunsche constant is a mathematical constant from the field of number theory . It is named after the mathematician Viggo Brun .

Brun's constant for prime twins

In 1919, the mathematician Viggo Brun showed that the sum of the reciprocal values ​​of all prime twins (pairs of prime numbers whose difference is 2) converges . The limit of this sum is called Brun's constant for prime twins: ${\displaystyle B_{2}}$

${\displaystyle B_{2}=\textstyle \!\sum \limits _{p,p+2\;{\text{prim}}}\!({\frac {1}{p}}+{\frac {1}{p+2}})=({\frac {1}{3}}+{\frac {1}{5}})+({\tfrac {1}{5}}+{\tfrac {1}{7}})+({\tfrac {1}{11}}+{\tfrac {1}{13}})+({\tfrac {1}{17}}+{\tfrac {1}{19}})+({\tfrac {1}{29}}+{\tfrac {1}{31}})+\ldots }$

This result of analytical number theory is surprising at first glance, since the sum of the reciprocal values ​​of all prime numbers diverges , as was proven by Leonhard Euler in the 18th century . If it were also divergent, one would have proof of the still open conjecture that there are an infinite number of prime twins ( Alphonse de Polignac (1817–1890) 1849). However, the convergence does not suggest the opposite. ${\displaystyle B_{2}}$

calculation

The idea for the calculation is that the summation is first carried out as far as possible and then the missing remainder is estimated. So have Daniel Shanks and John William Wrench, Jr. (1911-2009) all twin primes below 2 × 10 ⁶ used.

An estimate

${\displaystyle B_{2}\approx 1{,}90216\ 05831\ 04}$(Follow A065421 in OEIS )

comes from Pascal Sebah in 2002, who considered all prime twins up to 10 ^{16 for} this. However, the calculation of is extremely difficult, on the one hand, because the series converges very slowly, and on the other, because finding all large twins of prime numbers is extremely complicated ( see also: Primality tests ). ${\displaystyle B_{2}}$

The most accurate estimate so far is (as of March 16, 2010)

${\displaystyle B_{2}=1{,}90216\ 05832\ 09\pm 0{,}00000\ 00007\ 81.}$

For this, the reciprocal values ​​of all 19,831,847,025,792 prime twins below 2 · 10 ^{16 were} summed up:

${\displaystyle \textstyle \sum \limits _{p,p+2\;{\text{prim}} \atop <2\cdot 10^{16}}({\frac {1}{p}}+{\frac {1}{p+2}})=1{,}83180\ 80634\ 32379\ 01198\ 41239\ 12086\ 74712\ 537\ldots }$

and the remainder is estimated.

Brun's constant for prime number triplets

In addition there are two other Brun's constant and for prime triplets . ${\displaystyle B_{2}}$ ${\displaystyle B_{3a}}$ ${\displaystyle B_{3b}}$

The first three prime triplets of the form are (5, 7, 11), (11, 13, 17), and (17, 19, 23). In this case, too, the sum converges and the following applies (as of March 16, 2010): ${\displaystyle (p,p+2,p+6)}$

${\displaystyle B_{3a}=\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}\right)+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}\right)+\left({\tfrac {1}{17}}+{\tfrac {1}{19}}+{\tfrac {1}{23}}\right)+\ldots \approx 1{,}09785\ 10396\ 79}$

The first three prime triplets of the form are (7, 11, 13), (13, 17, 19), and (37, 41, 43). The sum also converges and the following applies (as of March 16, 2010): ${\displaystyle (p,p+4,p+6)}$

${\displaystyle B_{3b}=\left({\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)+\left({\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\left({\tfrac {1}{37}}+{\tfrac {1}{41}}+{\tfrac {1}{43}}\right)+\ldots \approx 0{,}83711\ 32124\ 11}$

Brun's constant for prime quadruplets

In addition , there is the Brun's constant for prime quadruplets , pairs of twin primes, which have a distance of 4 (this is the smallest possible distance between two twin primes each other). The first three prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19) and (101, 103, 107, 109), so ${\displaystyle B_{2}}$ ${\displaystyle B_{4}}$

${\displaystyle B_{4}=\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\left({\tfrac {1}{101}}+{\tfrac {1}{103}}+{\tfrac {1}{107}}+{\tfrac {1}{109}}\right)+\ldots }$

Since all summands of also occur in and except for and no summands are duplicated, this series also converges. She has the value (as of March 16, 2010) ${\displaystyle B_{4}}$ ${\displaystyle B_{2}}$ ${\displaystyle {\tfrac {1}{11}}}$ ${\displaystyle {\tfrac {1}{13}}}$

${\displaystyle B_{4}=0{,}87058\ 83799\ 75\pm 0{,}00000\ 00001\ 14}$(Follow A213007 in OEIS ).

Trivia

• In 1994 Thomas R. Nicely discovered the so-called Pentium FDIV bug while estimating over all prime twins up to 10 ¹⁴ . ${\displaystyle B_{2}}$
• Occasionally, the statement about the convergence of the sum of the reciprocals of all prime twins (i.e. the existence and calculability of Brun's constants) is called a Brunscher joke . The mathematical joke lies in the fact that, despite Brun's precise result, the really interesting question of whether there are infinitely many prime twins remains open (and the affirmative conjecture has not yet been proven).

literature

• Viggo Brun : La série où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie ${\displaystyle \textstyle {\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+{\frac {1}{19}}+{\frac {1}{29}}+{\frac {1}{31}}+{\frac {1}{41}}+{\frac {1}{43}}+{\frac {1}{59}}+{\frac {1}{61}}+\ldots }$ , Bulletin des Sciences Mathématiques 43, 1919, pp. 100–104, 124–128 (French; Gallica: gallica.bnf.fr )

Web links

• Eric W. Weisstein : Brun's Constant . In: MathWorld (English).
• Thomas R. Nicely: Prime Constellations Research Project .

Individual evidence

1. ↑ ^{a b c d} Thomas R. Nicely: Prime Constellations Research Project . March 16, 2010