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# Brun's constant

Brun's constant ${\displaystyle \scriptstyle B_{2}\,}$, named after Viggo Brun, is defined as the sum of the reciprocals of each prime belonging to twin prime pairs, where 5 belongs to the two twin prime pairs (3,5) and (5,7) and thus contributes twice. He proved the convergence of the series

${\displaystyle B_{2}={\bigg (}{\frac {1}{3}}+{\frac {1}{5}}{\bigg )}+{\bigg (}{\frac {1}{5}}+{\frac {1}{7}}{\bigg )}+{\bigg (}{\frac {1}{11}}+{\frac {1}{13}}{\bigg )}+{\bigg (}{\frac {1}{17}}+{\frac {1}{19}}{\bigg )}+{\bigg (}{\frac {1}{29}}+{\frac {1}{31}}{\bigg )}+{\bigg (}{\frac {1}{41}}+{\frac {1}{43}}{\bigg )}+{\bigg (}{\frac {1}{59}}+{\frac {1}{61}}{\bigg )}+{\bigg (}{\frac {1}{71}}+{\frac {1}{73}}{\bigg )}+{\bigg (}{\frac {1}{101}}+{\frac {1}{103}}{\bigg )}+{\bigg (}{\frac {1}{107}}+{\frac {1}{109}}{\bigg )}+\ldots \,}$

Note that the first twin prime pair is the only one that is not of the form ${\displaystyle \scriptstyle (6k-1,\,6k+1)\,}$. It is conjectured that there are an infinity of twin prime pairs, this is known as the twin prime conjecture.

## Decimal expansion of Brun's constant

The decimal expansion of Brun's constant is

${\displaystyle B_{2}=1.902160583104\ldots \,}$

giving the sequence of decimal digits (Cf. A065421)

{1, 9, 0, 2, 1, 6, 0, 5, 8, 3, 1, 0, 4, ...}

For some constants, we can give thousands or even millions of decimal places. And for some constants we can barely give a dozen places, if that. This is an example of the latter, since it converges extremely slowly. For the few places that we do know, we have at least three different people to thank: Robert G. Wilson v, Neil Sloane and Pascal Sebah.

It seems (is that the case?) that the number of decimal places obtained is about the square of the natural logarithm of the upper bound of the range for which we consider the twin prime pairs. For example, the above 13 decimal places have been obtained by considering all twin prime pairs up to 10 16, where ${\displaystyle \scriptstyle (\log(10^{16}))^{2}\,=\,13.0077\ldots \,}$. Also note that those 13 decimal places where obtained by a clever extrapolation method (which assumes the truth of the twin prime conjecture,) whereas using direct estimation we would have to go up to 10 530 just to reach 1.9![1]

## Continued fraction for Brun's constant

The simple continued fraction for Brun's constant is

${\displaystyle B_{2}=1+{\cfrac {1}{1+{\cfrac {1}{9+{\cfrac {1}{4+{\cfrac {1}{1+\ddots }}}}}}}}\,}$

giving the sequence of partial quotients (Cf. A??????)

{1, 1, 9, 4, 1, 1, 8, 3, 4, 7, 1, 3, ...}