In 1919 the Norwegian mathematician Viggo Brun proved that the sum of the reciprocals of the twin primes (pair of prime numbers which differ by 2) converges to a finite value now known as Brun's constant (B_{2}). Unlike some constants related to divergent sums of reciprocals of integers, like the Euler-Mascheroni constant (harmonic series) and Mertens constant (prime numbers), which are respectively known to millions and thousands of digits, the Brun's constant is known to 9 or 10 digits only. The sum converges very slowly, so slowly that the sum will not reach the value 1.9 until all the reciprocals of the twin prime pairs up to 10^{530} are summed up. Thus, an indirect method is used, assuming the Twin Prime Conjecture is true:

Bwhere^{*}_{2}= B_{2(p)}+ 4*C_{2}/log(p)

B_{2(p)}= sum of the reciprocals of the twin prime pairs up to p

C_{2}= Twin primes constant (0.66016181584686957..)

B^{*}_{2}= Approximation to Brun's constant

B^{*}_{2} has been calculated for p up to 10^{15} and 10^{16} by Thomas R. Nicely (1999) and Pascal Sebah (2012), respectively:

http://www.trnicely.net/twins/twins2.html

http://numbers.computation.free.fr/Constants/Primes/twin.pdf

By the way, the famous Pentium bug was discovered by Dr. Nicely in 1994 when calculating B_{2} for p up to 10^{14}.

Let's now compute a the constant to a few digits on the WP-34S:

001 LBL A 018 x<>yThe most accurate value of B

002 STO 03 019 RCL L

003 2 020 x>? 03

004 +/- 021 SKIP 007

005 STO 02 022 ||

006 0 023 RCL L

007 STO 01 024 x<>y

008 + 025 1/x

009 RCL L 026 STO+ 01

010 DEC X 027 x<>y

011 NEXTP 028 BACK 018

012 ENTER^ 029 RCL 00

013 ENTER^ 030 RCL 03

014 NEXTP 031 LN

015 - 032 /

016 x<>? 02 033 RCL+ 01

017 BACK 008 034 END2.64064726339 STO 00 ; 4*C

_{2}10 A --> 2.02300901133 ; B

_{21}

RCL 01 --> 0.87619047619 ; (1/3 + 1/5) + (1/5 + 1/7)100 A -->

1.90439963329 ; B_{22}

RCL 01 --> 1.33099036572 ; (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... + (1/83 + 1/89)EEX 3 A -->

1.90030530861 ; B_{23}

RCL 01 --> 1.51803246356 ; (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + ... + (1/991 + 1/997)EEX 4 A -->

1.90359819122 ; B_{24}EEX 5 A -->

1.90216329186 ; B_{25}EEX 6 A -->

1.90191335333 ; B_{26}EEX 7 A -->

1.90218826322 ; B_{27}(about 15 minutes on the WP-34S emulator)

_{2}to date is

1.902160583104

from which the last two or three digits are uncertain.

Now let's find a couple of suitable approximations:

On squaring

1.902160583104we get

3.61821488391Notice the three digits after the decimal point resemble those of the golden-ratio. Let's add 2 to the built-in Phi constant:

1.61803398875Let's divide the value obtained earlier by this one:

2 + -->

1.61803398875

/ --->Not bad! But there's more. Let's take the square of B

1.00004999819

_{2}again

3.61821488391and divide it by the golden-ratio:

1.61803398875Notice this is close to the square root of 5. So, let's square it:

/ -->

2.23617977686

2.23617977686One 9 better! These allow for the following nice approximations:

ENTER * -->

5.00049999444

Band_{2}~ Sqrt(1.00005*(Phi + 2)) = 1.90216058482

B_{2}~ Sqrt(Phi*Sqrt(5.0005)) = 1.90216058363

The latter is a great mnemonic aid: Phi, a constant related to the square root of 5, and a 5-digit number, beginning and ending with 5 and zeroes in the middle.

P.S.: No bugs were found on the WP-34S when running the above program :-)

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P.S.: Some optimization:

001 LBL A 019 x!=? 02

002 STO 04 020 BACK 007

003 # 048 021 x<> L

004 SDR 002 022 x>? 04

005 °->G 023 SKIP 007

006 STO 01 024 ||

007 2 025 y<> L

008 +/- 026 1/x

009 STO 02 027 STO+ 01

010 5 028 x<>y

011 + 029 RCL+ 03

012 STO 03 030 BACK 015

013 RCL L 031 RCL 00

014 DEC X 032 RCL 04

015 NEXTP 033 LN

016 FILL 034 /

017 NEXTP 035 RCL+ 01

018 - 036 END2.64064726339 STO 00

Emulador @ 1.86 GHz:

10^5: 1.90216329186 ( 6.1 s)

10^6: 1.90191335333 ( 63.1 s)

10^7: 1.90218826322 ( 759.6 s)

2*10^7: 1.90217962170 (1692.3 s)

_{Edited to add P.S.}

*Edited: 1 Jan 2013, 7:19 p.m. after one or more responses were posted*