Innumber theory,Brun's theorem states that the sum of thereciprocals of thetwin primes (pairs ofprime numbers which differ by 2)converges to a finite value known asBrun's constant, usually denoted byB₂ (sequenceA065421 in theOEIS). Brun's theorem wasproved byViggo Brun in 1919, and it has historical importance in the introduction ofsieve methods.

The convergence of the sum of reciprocals of twin primes follows from bounds on thedensity of the sequence of twin primes.Let${\displaystyle {\pi }_2(x)}$ denote the number ofprimesp ≤x for whichp + 2 is also prime (i.e.${\displaystyle {\pi }_2(x)}$ is the number of twin primes with the smaller at mostx). Then, we have

${\displaystyle {\pi }_2(x)=O\left(\frac{x(\log \log x{)}^2}{(\log x{)}^2}\right).}$

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 asmall set. In explicit terms, the sum

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

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 anirrational 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%.^[1]

By calculating the twin primes up to 10¹⁴ (and discovering thePentium FDIV bug along the way), Nicely heuristically estimated Brun's constant to be 1.902160578.^[1] Nicely has extended his computation to 1.6×10¹⁵ as of 18 January 2010 but this is not the largest computation of its type.

In 2002,Pascal Sebah andPatrick Demichel used all twin primes up to 10¹⁶ to give the estimate^[2] thatB₂ ≈ 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 Dominic Klyve showed conditionally (in an unpublished thesis) thatB₂ < 2.1754 (assuming theextended Riemann hypothesis). Then in 2025, Lachlan Dunn showedB₂ < 2.1609, assuming the generalised Riemann hypothesis.^[3] It has been shown unconditionally thatB₂ < 2.347.^[4]

There is also aBrun's constant for prime quadruplets. Aprime 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 byB₄, is the sum of the reciprocals of all prime quadruplets:

${\displaystyle B_4=\left(\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}\right)+\left(\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}\right)+\left(\frac{1}{101}+\frac{1}{103}+\frac{1}{107}+\frac{1}{109}\right)+\cdots }$

with value:

B₄ = 0.87058 83800 ± 0.00000 00005, the error range having a 99% confidence level according to Nicely.^[1]

This constant should not be confused with theBrun's constant forcousin primes, as prime pairs of the form (p, p + 4), which is also written asB₄. Wolf derived an estimate for the Brun-type sumsB_n of 4/n.

Let${\displaystyle C_2=0.6601\dots }$ (sequenceA005597 in theOEIS) be thetwin prime constant. Then it isconjectured that

${\displaystyle {\pi }_2(x)\sim 2C_2\frac{x}{(\log x{)}^2}.}$

In particular,

for every${\displaystyle \epsilon >0}$ and all sufficiently largex.

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

The digits of Brun's constant were used in a bid of $1,902,160,540 in theNortel patent auction. The bid was posted byGoogle and was one of three Google bids based onmathematical constants.^[5] Furthermore, academic research on the constant ultimately resulted in thePentium FDIV bug becoming a notablepublic relations fiasco forIntel.^[6][7]

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

• Weisstein, Eric W."Brun's Constant".MathWorld.
• Weisstein, Eric W."Brun's Theorem".MathWorld.
• Brun's constant atPlanetMath.
• Sebah, Pascal and Xavier Gourdon,Introduction to twin primes and Brun's constant computation, 2002. A modern detailed examination.
• Wolf's article on Brun-type sums

• Sieve theory
• Theorems about prime numbers

• Articles with short description
• Short description matches Wikidata
• CS1 French-language sources (fr)