Today's number is Brun's constant
It is well known that this series:
1 + 1/2 + 1/3 + 1/4 + 1/5 + ...
(the harmonic series) diverges. In 1737, Euler showed that the sum of the reciprocals of the primes also diverges:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
But let's now consider the reciprocals of the twin primes, where a twin prime pair is a pair of primes (p,p+2):
(1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17 + 1/19) + ...
Each twin prime pair contributes both reciprocals, so primes that belong to two twin prime pairs (such as 5) are counted twice.
By Brun's theorem, this series converges to a finite number. Remarkably, this is true whether there are finitely or infinitely many twin primes (an open problem).
In 1976, Richard P. Brent calculated all twin primes up to 100 billion and obtained an estimate for what this series converges to, namely:
B ≈ 1.9021605823, with an error of about 8 x 10^-10.
Since then, the number of terms has been calculated using twin primes up to 10^16, giving the result
B ≈ 1.902160583104
Despite more than a century of study, we still don't know whether there are infinitely many twin primes. Whatever the answer turns out to be, Brun's theorem remains one of the most surprising results in analytic number theory.
