Marvin Ray Burns

 I've been using Maple since 1997 or so.

MaplePrimes Activity


These are replies submitted by Marvin Ray Burns

For translating to Maple it is probably best to use Crandall's original code: 

 

(*Fastest (at RC's end) as of 30 Nov 2012.*)prec = 500000;(*Number of \

required decimals.*)ClearSystemCache[];
T0 = SessionTime[];
expM[pre_] := 
  Module[{a, d, s, k, bb, c, n, end, iprec, xvals, x, pc, cores = 4, 
    tsize = 2^7, chunksize, start = 1, ll, ctab, 
    pr = Floor[1.02 pre]}, chunksize = cores*tsize;
   n = Floor[1.32 pr];
   end = Ceiling[n/chunksize];
   Print["Iterations required: ", n];
   Print["end ", end];
   Print[end*chunksize];
   d = N[(3 + Sqrt[8])^n, pr + 10];
   d = Round[1/2 (d + 1/d)];
   {b, c, s} = {SetPrecision[-1, 1.1*n], -d, 0};
   iprec = Ceiling[pr/27];
   Do[xvals = Flatten[ParallelTable[Table[ll = start + j*tsize + l;
        x = N[E^(Log[ll]/(ll)), iprec];
        pc = iprec;
        While[pc < pr, pc = Min[3 pc, pr];
         x = SetPrecision[x, pc];
         y = x^ll - ll;
         x = x (1 - 2 y/((ll + 1) y + 2 ll ll));];(*N[Exp[Log[ll]/ll],
        pr]*)x, {l, 0, tsize - 1}], {j, 0, cores - 1}, 
       Method -> "EvaluationsPerKernel" -> 1]];
    ctab = Table[c = b - c;
      ll = start + l - 2;
      b *= 2 (ll + n) (ll - n)/((ll + 1) (2 ll + 1));
      c, {l, chunksize}];
    s += ctab.(xvals - 1);
    start += chunksize;
    Print["done iter ", k*chunksize, " ", SessionTime[] - T0];, {k, 0,
      end - 1}];
   N[-s/d, pr]];
 
t2 = Timing[MRBtest2 = expM[prec];];
MRBtest2 - MRBtest3
 
(*===============*).

Transcendental numbers that are not solitaire with respect to base 10:

Liouville's constant.

 

 

As you know the sin(x)~=x for small x, so instead of (10 Mantissa[sin(10^(-100 - 1/x))])^x we are really confronted with (10 Mantissa[10^(-100 - 1/x)])^x.= (10 Mantissa[10^(- 1/x)])^x = 10^(x-1). Thus we have the roots of the powers of 10 that I was concerned about. If I missed anything that you notice let me know.

You can replace the "1/x" with "any rational number/x" if you also replace the "^x" with "^one of the right rational numbers*x."

 

x        (10 Mantissa[sin(10^(-100 -3.14159260000000000/x))])^(5000000*x)

1       1.*  10^4292037

2       1.*  10^4292037

3      1.*  10^14292037

etc.

Here we use the mantissa.

Noticing that 3.162277660^2~=10 we have here a more subtle and beautiful pattern for sin(10^-k), using sufficiently large integral value for k; here we use 100 but 9 is usually sufficient.

 

 

x        (10 Mantissa[sin(10^(-100 - 1/x))])^x

1       1.*  10^1

2       1.*  10^1

3       1.* 10^2

4       1.* 10^3

5       1.* 10^4

6       1.* 10^5

etc.

 

x        (10 Mantissa[sin(10^(-100 - 2/x))])^x

1        1.* 10^0

2        1.* 10^2

3       1.* 10^1

4       1.* 10^2

5       1.* 10^3

6       1.* 10^4

etc.

 

x        (10 Mantissa[sin(10^(-100 - 3/x))])^x

1       1.*  10^1

2       1.*  10^1

3      1.*  10^3

4      1.*  10^1

5      1.*  10^2

6      1.*  10^3

etc.

 

x        (10 Mantissa[sin(10^(-100 - 4/x))])^x

1       1.*  10^1

2       1.*  10^0

3      1.*  10^2

4      1.*  10^4

5      1.*  10^1

6      1.*  10^2

7      1.*  10^3

etc.

 

x        (10 Mantissa[sin(10^(-100 - 5/x))])^x

1       1.*  10^1

2       1.*  10^1

3      1.*  10^1

4      1.*  10^3

5      1.*  10^4

6      1.*  10^1

7      1.*  10^2

8      1.*  10^3

etc.

 

The Mathematica code for this is 

Table[{x, (10 MantissaExponent[N[Sin[10^(-100 - 1/x)], 10]][[1]])^  x}, {x, 1, 10}] // TableForm

Change 1/x to 2/x,3/x, etc .

Can anyone figure out the pattern here?

Replacing 1/x with (3/2)/x  and ^x to ^(2x) we find

x        (10 Mantissa[sin(10^(-100 - (3/2)/x))])^(2x)

1       1.*  10^1

2      1.*   10^1

3      1.*  10^3

4      1.*  10^5

5      1.*  10^7

6      1.*  10^9

etc.

 

Replacing 1/x with (5/2)/x  and ^x to ^(2x) we find

x        (10 Mantissa[sin(10^(-100 - (5/2)/x))])^(2x)

1      1.*   10^1

2      1.*   10^3

3      1.*  10^1

4      1.*  10^3

5     1.*   10^5

6     1.*   10^7

etc.

 

Replacing 1/x with (5/3)/x  and ^x to ^(3x) we find

x        (10 Mantissa[sin(10^(-100 - (5/3)/x))])^(3x)

1      1.*   10^1

2      1.*   10^1

3      1.*  10^4

4      1.*  10^7

5      1.*  10^10

6      1.*  10^12

etc.

 

Another partial repeating decimal sequece:

n=22,25,28,30,31,33,34,36,37,39,40,42,43,45,46,48,49,51,52,54,55,57,58,60,61,

63,64,66,67,69,70,72,73,75,76,78,79,81,82,84,85,87,88,90,91,93,94,96,97,99,100 and some others 

give 41991753102864213975325086436197547308658.

 

Two more repeating decimal sequences are

864213975325086436197547308658419769530880641991753102

and

038965150076261187372298483409594520705631816742927854

There is at least one more that has the digits 5085671481 in it.

The Taylor series of sin(x) about x=0 is 

x-x^3/6+x^5/120-x^7/5040+x^9/362880-x^11/39916800+x^13/6227020800-x^
15/1307674368000+x^17/355687428096000-x^19/121645100408832000+O(x^20)

Can anyone describe the pattern here?

 

marvinrayburns.com

 

Let c be the Hafner-Sarnak-McCurley Constant = 0.3532363718549959845435165504326820112801... and g= Gamma[Pi] = 2.28803779534003241795958890906023392289...

m/g - (30780 c - 10003)/(7 (37 c + 1500)) = -7.99658771873458...*10^-21

 

I wish I knew how Wolfram Alpha does it! 

 

I tried reposting the table in my comment but it didn't work. So I just made the corrections in the table

 Here a sample Mathematica code I used:

WolframAlpha["7^6^5^4^3^2", {{"LastFewDigits", 1}, "Plaintext"}]

 

It returned

"...9058585601"

Good catch. The ones with only 9 digits have a missing first or last displayed digit. I replaced the missing digits.

I computed a little over 1,200,000 digits of the MRB constant in 11 days, 21 hours, 17 minutes, and 41 seconds.

 

 

 

Ricard Crandall computed 1,048,576 digits in a lighting fast 76.4 hours.

Here are my best timings of computing digits of the MRB constant using Crandall's method in seconds:



Digits 1.7GH i5 (laptop) 3.5 GH AMD 2 GH(Xeon) ECC RAM 3.2 GH .i7 extreme

 3.2 GH .i7 extreme (Method optimized for large computations)

500 0.18 0.15 ? 0.062 0.086 
1000 0.38 0.35 0.265 0.142 0.1230
2000 1.01 0.88 0.67 0.48 0.4050
3000 1.99 1.7 1.28 1.01 0.8640
4000 4.22 3.08 2.55 1.62 1.5290 
5000 6.32 5.03 3.72 2.65 2.4801
6000 9.46 7.67 5.3 3.97 3.6412
7000 14.32 10.84 7.94 5.66 5.0272
8000 18.9 14.05 10.32 7.8 5.9803
9000 23.87 18.01 12.96 9.94 7.7264
10000 28.96 23.35 17.17 12.32 9.8515
20000 187 110 83 63 48.0657
30000 449 304 231 174 121.3639
40000 1407 607 440 349 236.8525
50000 ? 1041 736 563 366.2109
60000 ? 1676 1224 953 607.8767
100000 ? 6928 5487 4054 2655.2108
300000 ? 84428 61200 44994 31641
500000 ? 243903 ? ?  
1000000 ? ? ? 686377  
1200000 ? ? ? 1027061  
           
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