“Brun's theorem”的意思、由来-开放百科全书

In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B₂ {{OEIS|id=A065421}}. Brun's theorem was proved by Viggo Brun in 1919, and it has historical importance in the introduction of sieve methods.

Asymptotic bounds on twin primes

The convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes.

Let

denote the number of primes p ≤ x for which p + 2 is also prime (i.e.

is the number of twin primes with the smaller at most x). Then, for x ≥ 3, we have

That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor.

It follows from this bound that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form a small set. In explicit terms the sum

either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant.

The fact that the sum of the reciprocals of the prime numbers diverges implies that there are infinitely many prime numbers. Because the sum of the reciprocals of the twin primes instead converges, it is not possible to conclude from this result that there are finitely many or infinitely many twin primes. Brun's constant could be an irrational number only if there are infinitely many twin primes.

Numerical estimates

The series converges extremely slowly. Thomas Nicely remarks that after summing the first one billion (10⁹) terms, the relative error is still more than 5%.^[1].

By calculating the twin primes up to 10¹⁴ (and discovering the Pentium FDIV bug along the way), Nicely heuristically estimated Brun's constant to be 1.902160578.^[2] Nicely has extended his computation to 1.6{{e|15}} as of 18 January 2010 but this is not the largest computation of its type.

In 2002, Pascal Sebah and Patrick Demichel used all twin primes up to 10¹⁶ to give the estimate:^[3]

It is based on extrapolation from the sum 1.830484424658... for the twin primes below 10¹⁶. Dominic Klyve showed conditionally (in an unpublished thesis) that B₂ < 2.1754 (assuming the extended Riemann hypothesis). It has been shown unconditionally that B₂ < 2.347.^[4]

There is also a Brun's constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B₄, is the sum of the reciprocals of all prime quadruplets:

This constant should not be confused with the Brun's constant for cousin primes, as prime pairs of the form (p, p + 4), which is also written as B₄. Wolf derived an estimate for the Brun-type sums B_n of 4/n.

Further results

Many special cases of the above have been proved. Most recently, Jie Wu proved that for sufficiently large x,

In popular culture

The digits of Brun's constant were used in a bid of $1,902,160,540 in the Nortel patent auction. The bid was posted by Google and was one of three Google bids based on mathematical constants.^[6]

See also

Notes

1. ^{{cite web |url=http://www.trnicely.net/twins/twins2.html |last=Nicely |first=Thomas R. |title=Enumeration to 1.6*10^15 of the twin primes and Brun's constant |accessdate=16 February 2010 |work=Some Results of Computational Research in Prime Numbers (Computational Number Theory) |date=18 January 2010}}
2. ^{{cite web |url=http://www.trnicely.net/twins/twins2.html |last=Nicely |first=Thomas R. |title=Enumeration to 1.6*10^15 of the twin primes and Brun's constant |accessdate=16 February 2010 |work=Some Results of Computational Research in Prime Numbers (Computational Number Theory) |date=18 January 2010}}
3. ^{{cite web | url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.464.1118&rep=rep1&type=pdf | last1=Sebah | first1=Pascal | last2=Gourdon | first2=Xavier | title=Introduction to twin primes and Brun’s constant computation | access-date=5 January 2018}}
4. ^{{cite web |url=http://libarchive.dartmouth.edu/cdm/ref/collection/dcdis/id/36604 |last=Klyve |first=Dominic |title=Explicit bounds on twin primes and Brun's Constant |accessdate=13 May 2015}}
5. ^{{cite web |url=http://www.trnicely.net/quads/quads.html |last=Nicely |first=Thomas R. |title=Enumeration to 1.6{{e|15}} of the prime quadruplets |accessdate=9 March 2009 |work=Some Results of Computational Research in Prime Numbers (Computational Number Theory) |date=26 August 2008}}
6. ^{{cite web | url=https://www.reuters.com/article/2011/07/02/us-dealtalk-nortel-google-idUSTRE76104L20110702 | title=Dealtalk: Google bid "pi" for Nortel patents and lost | publisher=Reuters | date=1 July 2011 | accessdate=6 July 2011 | author=Damouni, Nadia}}