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## Computing Digits of the MRB Constant

Maple

The MRB constant is defined at http://mathworld.wolfram.com/MRBConstant.html.

On about Dec 31, 1998 I computed 1 digit of the MRB constant with my TI-92's, by adding 1-sqrt(2)+3^(1/3)-4^(1/4) as far as I could. That first digit by the way is just 0. On Jan 11, 1999 I computed 3 digits of the MRB constant with the Inverse Symbolic Calculator.

In Jan of 1999 I computed 4 correct digits of the MRB constant using Mathcad 3.1 on a 50 MHz 80486 IBM 486 personal computer operating on Windows 95.

Shortly afterwards I computed 9 correct digits of the MRB constant using Mathcad 7 professional  on the Pentium II mentioned below.

On Jan 23, 1999 I computed 500 digits of the MRB constant with the online tool called Sigma.

Around September of 1999, I, computed the first 5,000 digits of the MRB Constant using Pari on a Pentium II IBM Aptiva 350 (CPU type 66MHz Intel i486DX2) Computer with 64 MB of ram. I  first computed 7000 digits on June 10-11, 2003 over a period, of 10 hours, on a PACKARD BELL 450mh P3 with an available 512MB RAM. Using a Sony Vaio P4 2.66 GHz laptop computer with 960 MB of available RAM I computed 8000 digits of the MRB constant, finishing at 2:04 PM on 3/25/2004.

The following computations in Mathematica 5.2 and 6.0 used a code similar to this:

f[mx_] := Block[{\$MaxExtraPrecision = mx + 8, a, b = -1, c = -1 - d, d = (3 + Sqrt)^n, n = 131 Ceiling[mx/100], s = 0}, a = 1; d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++]; For[k = 0, k < n, c = b - c; b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++]; N[1/2 - s/d, mx]]; RealDigits[ f, 10][]

On March 01, 2006 with a Gateway Media Center Desktop PD 3GHz with 2GB RAM available, computed the first 11,000 digits of the MRB Constant.

On Nov 24, 2006 I computed 40, 000 digits of the MRB Constant in 33hours and 26min with Mathematica 5.2.

(This computation was run on the previously mentioned 32-bit Windows 3GH PD desktop computer using 3.25 GB of Ram.)

I computed 50,000 digits of MRB Constant in 31 hours on a 2.6 GH AMD 64 bit Athlon laptop.

Max memory used was 3.5 GB RAM.

Finishing on July 29, 2007 at 11:57 PM EST, I computed 60,000 digits of MRB Constant. (This was computed in 50.51 hours on a 2.6 GH AMD Athlon with 64 bit Windows XP. Max memory used was 4.0 GB of RAM.)

Finishing on Aug 3 , 2007 at 12:40 AM EST, I computed 65,000 digits of MRB Constant.

Computed in only 50.50 hours on a Dell 2.66GH Core2Duo using 64 bit Windows XP. Max memory used was 5.0 GB of RAM.

Finishing on Aug 12, 2007 at 8:00 PM EST, I computed 100,000 digits of MRB Constant.

Computed in 170 hours on a 2.66GH Core2Duo using 64 bit Windows XP.

Max memory used was 11.3 GB of RAM. Median memory used was 8.5 GB of RAM.

Finishing on Sep 23, 2007 at 11:00 AM EST, I computed 150,000 digits of MRB Constant.

(This was computed in 330 hours on a 2.66GH Core2Duo using 64 bit Windows XP.

Max memory used was 22 GB of RAM. Median memory used was 17 GB of RAM.)

Finishing on March 16, 2008 at 3:00 PM EST, I computed 200,000 digits of MRB Constant.

(This was computed in 845 hours on a 2.66GH Core2Duo using 64 bit Windows XP.

Max memory used was 47 GB of RAM. Median memory used was 28 GB of RAM.)

On September 18, 2008 I started a computation of 225,000 digits of MRB Constant with a 2.66GH Core2Duo using 64 bit Windows XP. It was completed in a mere 1072 hours.

I wonder if anyone cares to compare the following computation of 260,000 digits with any you can do with Maple or perhaps a low level programing language?

The first 260,000 digits, to the right of the decimal place, of the MRB constant are now at http://marvinrayburns.com/MRB260K.txt.

This computation of 260,000 digits began on  "Thu 17 Jun 2010 19:33:43" and finished on "Wed 14 Jul 2010 04:27:17."
That is a total of 644.9 hours. Within that time, the computation took up 630.306 hours of processor time on an Dell Studio XPS 8100 i7 860 2.80 GH 2.80 GH with 8GB physical DDR3 RAM. The computer reserved an additional 39 GB virtual RAM for the one process of computing 260,000 digits. The Mathematica V7 command used was as follows:
DateString[]
a = AbsoluteTime[]
NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]},
WorkingPrecision -> 260000, Method -> "AlternatingSigns"] // Timing
AbsoluteTime[] - a
DateString[].

For verification,
the first 250,000 digits are the same as found in http://marvinrayburns.com/250KMRB.txt.
The 250,000 were computed by a 64 bit XP Dell, Optiplex GX745  2.66 GH using 4GB DDR2 Ram on board and 36 GB virtual, taking 333.102 hours of processor time, finishing on Jan 29, 2009, 1:26:19 pm (UTC-0500) EST.

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