View source for Brun theorem

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''on prime twins''

The series $\sum 1/p$ is convergent if $p$ runs through all (the first members of all) prime [[Twins|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 [[#References|[1]]]. The convergence of a similar series for generalized twins was proved at a later date.

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

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Trost,   "Primzahlen" , Birkhäuser  (1953)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Halberstam,   H.-E. Richert,   "Sieve methods" , Acad. Press  (1974)</TD></TR>
<TR><TD valign="top">[b1]</TD> <TD valign="top"> Steven R. Finch, ''Mathematical Constants'', Cambridge University Press (2003) ISBN 0-521-81805-2  {{ZBL|1054.00001}}</TD></TR>
</table>
 	
[[Category:Number theory]]