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# Template:Sequence of the Day for August 15

Intended for: August 15, 2011

## Timetable

• First draft entered by Alonso del Arte on May 30, 2011 (as a verbatim copy of a write-up from December 15, 2010) ✓
• Draft reviewed by Alonso del Arte on August 12, 2011
• Draft approved by Daniel Forgues on August 12, 2011

The line below marks the end of the <noinclude> ... </noinclude> section.

A065421: Decimal expansion of the twin primes Brun’s constant
 B2
:
 ∑
 1 p
+
 1 p + 2
as
 (  p, p + 2)
runs through the twin prime pairs.
 1 3
+
 1 5
+
 1 5
+
 1 7
+
 1 11
+
 1 13
+
where the first twin prime pair is the only one that is not of the form
 (6 k  −  1, 6 k  +  1), k   ≥   1
.
1.902160583104...

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. Today’s Sequence of the Day 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 (log (10 16 )) 2 = 13.0077... 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! (Sebah and Gourdon)

_______________

* Pascal Sebah and Xavier Gourdon, Introduction to twin primes and Brun's constant computation, July 30, 2002.
* Weisstein, Eric W., Brun's Constant, from MathWorld—A Wolfram Web Resource.