# Brun theorem

From Encyclopedia of Mathematics

*on prime twins*

The series $\sum 1/p$ is convergent if $p$ runs through all (the first members of all) prime twins. This means that even if the number of prime twins is infinitely large, they are still located in the natural sequence rather sparsely. This theorem was demonstrated by V. Brun [1]. The convergence of a similar series for generalized twins was proved at a later date.

#### References

[1] | V. Brun, "La série $\frac1{5} + \frac1{7} + \frac1{11} + \frac1{13} + \frac1{17} + \frac1{19} + \frac1{29} + \frac1{31} + \frac1{41} + \frac1{43} + \frac1{59} + \frac1{61} + \ldots$ où les dénominateurs sont "nombres premiers jumeaux" est convergente ou finie" Bull. Sci. Math. (2) , 43 (1919) pp. 100–104; 124–128 |

[2] | E. Trost, "Primzahlen" , Birkhäuser (1953) |

#### Comments

#### References

[a1] | H. Halberstam, H.-E. Richert, "Sieve methods" , Acad. Press (1974) |

#### Comments

The value of the sum over all elements of prime twins has been estimated as 1.9021605831….

#### References

[b1] | Steven R. Finch, Mathematical Constants, Cambridge University Press (2003) ISBN 0-521-81805-2 Zbl 1054.00001 |

**How to Cite This Entry:**

Brun theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Brun_theorem&oldid=51647

This article was adapted from an original article by N.I. Klimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article