Brun's Theorem
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Theorem
The sum of the reciprocals of all the twin primes converges.
Proof
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Also see
Source of Name
This entry was named for Viggo Brun.
Historical Note
While Brun's Theorem proves that the sum of the reciprocals of all the twin primes converges, it has not yet been established what it actually converges to.
The number, whatever it is, is referred to as Brun's constant. Its best estimate is updated as and when the occasion demands.
As pointed out by David Wells in his Curious and Interesting Numbers of $1986$:
- Its value is exceedingly hard to calculate.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 90195 \ldots$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 90216 \, 054 \ldots$