# Sum of Reciprocals of Twin Primes

Jump to navigation
Jump to search

## 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.

This article, or a section of it, needs explaining.In particular: How would it be possible for the sum to be convergent but not actually finite? Explanation needed.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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