The number obtained by adding the reciprocals of the odd twin
primes,
(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 all prime
reciprocals 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).
Shanks and Wrench (1974) used all the twin primes among the first 2 million numbers. Brent (1976) calculated all twin
primes up to 100 billion and obtained (Ribenboim 1989, p. 146)
(Cipra 1995, 1996), in the process discovering a bug in Intel's® PentiumTM microprocessor. Using twin primes up to , Nicely (2000) subsequently obtained the result
(4)
The number of terms has since been calculated using twin primes up to
(Sebah 2002), giving the result
(5)
(OEIS A065421). Note that the value for given by Le Lionnais (1983) is incorrect.
Segal (1930) proved that Brun-type sums of over consecutive primes separated by are convergent (Halberstam and Richert 1983, p. 92).
Wolf suggests that
is roughly equal to
which, in the
case of twin primes, gives instead of .... Wolf also considers the "cousin
primes" Brun's constant .
(which provides a stronger characterization of the
divergence than Euler's proof that 
, obtained more than a century before Mertens'
proof).
, Nicely (1996) obtained
, Nicely (2000) subsequently obtained the result
(Sebah 2002), giving the result
given by Le Lionnais (1983) is incorrect.
of 
over consecutive primes separated by 
are convergent (Halberstam and Richert 1983, p. 92).
Wolf suggests that 
is roughly equal to 
which, in the 
case of twin primes, gives 
instead of 
.... Wolf also considers the "cousin
primes" Brun's constant 
.
."
Math. Comput. 30, 379, 1976.Brun, V. "La serie 
,
les dénominateurs sont nombres premiers jumeaux est convergente où
finie." Bull. Sci. Math. 43, 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, 1996.