Sum of Reciprocals of Twin Primes

Theorem

The sum of the reciprocals of all the twin primes:

$\dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 {11} + \dfrac 1 {13} + \dfrac 1 {17} + \dfrac 1 {19} + \dfrac 1 {29} + \dfrac 1 {31} + \cdots$

is either finite or convergent.

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Proof

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Historical Note

The Sum of Reciprocals of Twin Primes was established to be either finite or convergent in $1921$ by Viggo Brun.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes