Brun's constant, usually denoted or , is the sum of the reciprocals
of both members of every pair of twin primes. If is the set of primes for which is also a prime number,
then
(1)
By Brun's theorem, the series converges to a definite number, which expresses the scarcity of twin
primes, even if there are infinitely many of them (Ribenboim 1989, p. 201).
By contrast, the series of the reciprocals
of all primes diverges to infinity, as follows from
the Mertens second theorem by letting (which provides a stronger
characterization of the divergence than Euler's proof that , obtained more than a century before Mertens'
proof).
The extrapolated numerical estimates below rely on heuristic assumptions. Their quoted uncertainties are not rigorous error bounds (Nicely 1995, 2001).
Shanks and Wrench (1974) used all the twin primes up to 32,452,843 (Brent 1975, p. 53). Brent's tables, reviewed by Shanks (1976),
extended the computation to 100 billion and gave (Ribenboim 1989, p. 146)
were proved by Platt and Trudgian (2020). Assuming the generalized Riemann hypothesis, Dunn (2026) proved the stronger upper bound . Note that the value for given by Le Lionnais (1983) is incorrect.
For a fixed positive even number, define the Brun-type sum
(7)
The sum is indexed by pairs, so a prime number occurring in two pairs contributes twice. In particular, 5 occurs in the two twin
prime pairs
and .
Segal (1930) proved convergence of these sums (Halberstam and Richert 1974, p. 92).
Wolf (1997, p. 5) suggested the rough approximation , giving instead of .... His more detailed heuristic also depends on the prime factors of and contains an unknown error term. Wolf (1996) considered
the consecutive cousin prime pairs beginning with
. Including the nonconsecutive cousin
pair
would add
to that convention's value.
or 
, is the sum of the reciprocals
of both members of every pair of twin primes. If 
is the set of primes 
for which 
is also a prime number,
then
(which provides a stronger
characterization of the divergence than Euler's proof that 
, obtained more than a century before Mertens'
proof).
, Nicely (1995) obtained
, Nicely (2001) reported the result
(Sebah 2002), giving the estimate
. Note that the value for 
given by Le Lionnais (1983) is incorrect.
, define the Brun-type sum
and 
.
Segal (1930) proved convergence of these sums (Halberstam and Richert 1974, p. 92).
Wolf (1997, p. 5) suggested the rough approximation 
, giving 
instead of 
.... His more detailed heuristic also depends on the prime factors of 
and contains an unknown error term. Wolf (1996) considered
the consecutive cousin prime pairs beginning with
. Including the nonconsecutive cousin
pair 
would add 
to that convention's value.
,
les dénominateurs sont nombres premiers jumeaux est convergente ou finie."
Bull. Sci. Math. 43, 100-104 and 124-128, 1919.Cipra,
B. "How Number Theory Got the Best of the Pentium Chip." Science 267,
175, 1995.Cipra, B. "Divide and Conquer." 
of the Twin Primes and Brun's Constant." Virginia J. Sci. 46,
195-204, 1995. 
."
Math. Comput. 30, 379, 1976.