Brun's theorem

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

Let $\backslash pi\_2(x)$ denote the number of primes p ≤ x for which p + 2 is also prime (i.e. $\backslash pi\_2(x)$ is the number of twin primes with the smaller at most x). Then, we have

:$\backslash pi\_2(x)\; =\; O\backslash !\backslash left(\backslash frac\; \{x(\backslash log\backslash log\; x)^2\}\{(\backslash log\; x)^2\}\; \backslash right)\backslash !.$

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

This bound gives the intuition 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

:$\backslash sum\backslash limits\_\{\; p\; \backslash ,\; :\; \backslash ,\; p\; +\; 2\; \backslash in\; \backslash mathbb\{P\}\; \}\; \{\backslash left(\; \{\backslash frac\{1\}\{p\}\; +\; \backslash frac\{1\}\{\{p\; +\; 2\}\}\}\; \backslash right)\}\; =\; \backslash left(\; \{\backslash frac\{1\}\{3\}\; +\; \backslash frac\{1\}\{5\}\}\; \backslash right)\; +\; \backslash left(\; \{\backslash frac\{1\}\{5\}\; +\; \backslash frac\{1\}\{7\}\}\; \backslash right)\; +\; \backslash left(\; \{\backslash frac\{1\}\{\{11\}\}\; +\; \backslash frac\{1\}\{\{13\}\}\}\; \backslash right)\; +\; \backslash cdots$

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

If it were the case that the sum diverged, then that fact would imply that there are infinitely many twin primes. 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.

The series converges extremely slowly. Thomas Nicely remarks that after summing the first billion (10⁹) terms, the relative error is still more than 5%.{{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 |access-date=16 February 2010 |work=Some Results of Computational Research in Prime Numbers (Computational Number Theory) |date=18 January 2010 |archive-url=https://web.archive.org/web/20131208192242/http://trnicely.net/twins/twins2.html |archive-date=8 December 2013 |url-status=dead }}

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. 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{{cite CiteSeerX | last1=Sebah | first1=Pascal | last2=Gourdon | first2=Xavier | title=Introduction to twin primes and Brun's constant computation |citeseerx = 10.1.1.464.1118}} that B₂ ≈ 1.902160583104. Hence,

The last 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). Then in 2025, Lachlan Dunn showed B₂ < 2.1609, assuming the generalised Riemann hypothesis.{{cite arXiv |eprint=2504.15658 |last1=Dunn |first1=Lachlan |title=Improved Upper Bound on Brun's Constant Under GRH |date=2025 |class=math.NT }} It has been shown unconditionally that B₂ < 2.347.{{cite web |url=https://collections.dartmouth.edu/archive/object/dcdis/dcdis-klyve2007?ctx=dcdis#?length=12&start=0&view=list&rdat_only_u=no&rdat_u=yes&col=dcdis&oc_0=main-title&od_0=a&sv=Brun%27s+constant |last=Klyve |first=Dominic |title=Explicit bounds on twin primes and Brun's Constant |access-date=24 May 2021}}

{{anchor|Prime quadruplets}}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:

:$B\_4\; =\; \backslash left(\backslash frac\{1\}\{5\}\; +\; \backslash frac\{1\}\{7\}\; +\; \backslash frac\{1\}\{11\}\; +\; \backslash frac\{1\}\{13\}\backslash right)$

+ \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 value:

:B₄ = 0.87058 83800 ± 0.00000 00005, the error range having a 99% confidence level according to Nicely.

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.

Let $C\_2=0.6601\backslash ldots$ {{OEIS|A005597}} be the twin prime constant. Then it is conjectured that

:$\backslash pi\_2(x)\; \backslash sim\; 2C\_2\backslash frac\{x\}\{(\backslash log\; x)^2\}.$

In particular,

:

for every $\backslash varepsilon>0$ and all sufficiently large x.

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

:

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.{{cite news | url=https://www.reuters.com/article/us-dealtalk-nortel-google-idUSTRE76104L20110702 | title=Dealtalk: Google bid "pi" for Nortel patents and lost | work=Reuters | date=1 July 2011 | access-date=6 July 2011 | author=Damouni, Nadia | archive-date=3 July 2011 | archive-url=https://web.archive.org/web/20110703071724/http://www.reuters.com/article/2011/07/02/us-dealtalk-nortel-google-idUSTRE76104L20110702 | url-status=live }} Furthermore, academic research on the constant ultimately resulted in the Pentium FDIV bug becoming a notable public relations fiasco for Intel.{{cite web |url=http://www.trnicely.net/pentbug/pentbug.html |title=Pentium FDIV flaw FAQ |website=www.trnicely.net |access-date=22 February 2022 |archive-url=https://web.archive.org/web/20190618044444/http://www.trnicely.net/pentbug/pentbug.html |archive-date=18 June 2019 |url-status=dead}}{{Cite journal|url=https://ieeexplore.ieee.org/document/372360|doi = 10.1109/40.372360|title = Pentium FDIV flaw-lessons learned|year = 1995|last1 = Price|first1 = D.|journal = IEEE Micro|volume = 15|issue = 2|pages = 86–88}}

• Divergence of the sum of the reciprocals of the primes
• Meissel–Mertens constant

{{reflist}}

{{refbegin}}

• {{cite journal | first=Viggo|last= Brun | author-link=Viggo Brun | title=Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare | journal=Archiv for Mathematik og Naturvidenskab | volume=B34 | issue=8 | year=1915 }}
• {{cite journal|language=fr| first=Viggo|last= Brun | author-link=Viggo Brun | title=La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie | url=http://gallica.bnf.fr/ark:/12148/bpt6k486270d |journal=Bulletin des Sciences Mathématiques | year=1919 | volume=43 | pages=100–104, 124–128 }}
• {{cite book | first1=Alina Carmen|last1= Cojocaru|author1-link= Alina Carmen Cojocaru | first2=M. Ram|last2= Murty |author2-link=M. Ram Murty| title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=Cambridge University Press | isbn=0-521-61275-6 | pages=73–74 | year=2005 }}
• {{cite book | first=E.|last=Landau | author-link=Edmund Landau | title=Elementare Zahlentheorie | location=Leipzig, Germany | publisher=Hirzel | year=1927}} Reprinted Providence, RI: Amer. Math. Soc., 1990.
• {{cite book | first=William Judson|last= LeVeque |author-link=William J. LeVeque| title=Fundamentals of Number Theory | location=New York City| publisher=Dover Publishing | year=1996 | pages=1–288 | isbn=0-486-68906-9}} Contains a more modern proof.
• {{cite journal |first=J. |last= Wu |title=Chen's double sieve, Goldbach's conjecture and the twin prime problem |journal= Acta Arithmetica |volume=114 |pages=215–273 |year=2004 |orig-year=24 Sep 2007 |issue=3 |doi= 10.4064/aa114-3-2 |arxiv= 0705.1652|bibcode=2004AcAri.114..215W }}

{{refend}}

• {{mathworld|urlname=BrunsConstant|title=Brun's Constant}}
• {{mathworld|urlname=BrunsTheorem|title=Brun's Theorem}}
• {{PlanetMath|urlname=BrunsConstant|title=Brun's constant}}
• Sebah, Pascal and Xavier Gourdon, [http://numbers.computation.free.fr/Constants/Primes/twin.pdf Introduction to twin primes and Brun's constant computation], 2002. A modern detailed examination.
• [https://web.archive.org/web/20110109082911/http://www.ift.uni.wroc.pl/~mwolf/ Wolf's article on Brun-type sums]

Category:Sieve theory

Category:Theorems about prime numbers