The MRB constant = -1+.From the vast universe of (dd), where dd is defined as follows:
∞
∑
n=1
n
(-1)
1
n
n
In[]:=
TranscendentalRecognize[(num_)?NumericQ,(basis_)?VectorQ]:=Module[{lr,ans},lr=FindIntegerNullVector[Prepend[N[basis,Precision[num]],num]];ans=Rest[lr].basis/First[lr];Sign[N[ans]]*Sign[num]*ans]
S=N[NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision110],64];ST=NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision210];(*p=Table[n^(1/n),{n,2,64}];d=Table[Log[Prime[n]],{n,1,5}];e=Table[Exp[Prime[n]],{n,1,5}];s=Table[StieltjesGamma[n],{n,2,5}];*)r=Table[Pi^n,{n,2,67}];(*t=Table[Sum[x^(-n!),{n,2,Infinity}],{x,2,10}];*)dd=Union[(*t,p,1/p,d,1/d,s,1/s,e,1/e,...theseweretriedseperately*)r,1/r,{EulerGamma,GoldenRatio,GoldenAngle}];For[c=1,c<10,Block[{$MaxExtraPrecision=400},For[a=1,a<1000,result=TranscendentalRecognize[N[ST,110(*{28,64,100,110},"28"wasreplacedwitheachofthesesamplesizes*)],RandomSample[dd,5]];If[Abs[ST-Abs[result]]<10^-110,If[N[Abs[Denominator[result]],10]<10^4,Print["Simplest Expression: ",result];Print["Error: ",N[Abs[ST-Abs[result]],10]]]];a++]];c++]
1
0
Simplest Expression: ++++
1
1052
16224950379073492638548
65
π
36980076885177876820985
64
π
43203855726258231471856
59
π
33480119179867469551733
58
π
15200019234302372801633
40
π
Error: 1.188310169×
-118
10
1
0
Simplest Expression: ++++
1
1069
2407738790630506608574
65
π
2015737192125387195322
54
π
1374029826550259547760
52
π
3801113228818951221114
46
π
4915243062733563771080
39
π
Error: 4.169416914×
-115
10
Simplest Expression: +-++
1
9219
12961345739997508271386
60
π
92664515819308204320560
49
π
50668856943432214267606
45
π
10702818493932297555001
42
π
132280493401832496822985
40
π
Error: 5.596849286×
-121
10
Simplest Expression: +--+
1
1528
153496419640733578372215
65
π
19952671064595112265418
58
π
1519040746262392315381
57
π
236743424126773500718128
44
π
241884248163221298174462
42
π
Error: 4.528220703×
-123
10
1
0
Simplest Expression: -+++
1
34
816987238233752571822635601
67
π
1735404687960196266416183862
60
π
713674999906738904385570004
58
π
1423540456792124102683274409
53
π
Error: 1.356725427×
-111
10
Out[]=
$Aborted
...
In[]:=
2,000,000 optimal approximations of the 20 to 110 digits of MRB constant were made. By a vast margin, the 42 simplest found were of the following form, and were remarkably similar expressions equal to the MRB constant to more than 110 accurate digits.
Interestingly, all the inner denominators of these first-to-be found such rational-coefficient approximations to the MRB constant are of powers of to .
Interestingly, all the inner denominators of these first-to-be found such rational-coefficient approximations to the MRB constant are of powers of
π
from
36
π
67
π
Theclosestonewasoundtwice:
Simplest Expression: -++-+
1
722
2816430872233976175430372
62
π
5957097815670284202075483
60
π
4424218174017480708311859
59
π
8072099516513147854361561
56
π
3192409539544820132530437
45
π
Error: 6.798165174×
-134
10
Simplest Expression: -++-+
1
722
2816430872233976175430372
62
π
5957097815670284202075483
60
π
4424218174017480708311859
59
π
8072099516513147854361561
56
π
3192409539544820132530437
45
π
Error: 6.798165174×
-134
10
"Simplest Expression: "---+
1
9829
11780815272846921339491439
66
π
9454446634820560766932241
59
π
6933452473184130184014486
58
π
9608226003969361773975237
57
π
13833668461810598004782066
44
π
Error: 9.787694535×
-133
10
Simplest Expression: -++-+
1
2131
10294084256347200003846148
65
π
6967133162807652571107565
64
π
347545356962160341271162
59
π
25177427607075559412461700
57
π
9422420158868391595375929
45
π
Error: 3.046571730×
-133
10
Simplest Expression: -+--+
1
199
9714089915919409138406
61
π
12213839844938905338610
59
π
23861825046536702985423
54
π
818762804867886579582
52
π
9032997220721828869718
41
π
Error: 5.390762258×
-115
10
Simplest Expression: +-++
1
1969
574600728160998342739
58
π
977353715661272216646
57
π
2069322398693938769762
47
π
830702257795974686037
40
π
283540769574875165993
36
π
Error: 1.589711303×
-111
10
Simplest Expression: -++-+
1
2
274043676231736792233
67
π
10416401276131596360391
49
π
8331565086773668803864
48
π
2878373182826936594597
44
π
1774154322237604888022
43
π
Error: 2.578424152×
-112
10
Simplest Expression: ---++
1
95
8778896325705027821831
65
π
4511368312215809760412
64
π
11224473801108068422388
58
π
30666853951092463428648
49
π
13537115381946876676097
42
π
Error: 1.394066556×
-114
10
Simplest Expression: -++-+
1
72
5022709862568054252146
62
π
6915467884897425469086
58
π
8436036169477165554041
50
π
278074297231976057835
44
π
3276901420809908649936
41
π
Error: 1.077422413×
-114
10
Simplest Expression: ++++
1
760
3558484922814618996681
59
π
4517255303907734226981
53
π
360593113111658107712
52
π
1645449266612738417245
46
π
3494816246944647556926
39
π
Error: 3.935290151×
-113
10
Simplest Expression: -++-+
1
62
2463323831216348319806
66
π
13758274660583294926600
60
π
12108578488664280230316
59
π
1941155664326074081565
50
π
2814362319206579319356
41
π
Error: 9.872282059×
-115
10
Simplest Expression: ++-+
1
6161
2051524408851399572662
54
π
58336715524973494483
48
π
806585706547559366569
43
π
1743316223389646804045
42
π
915407153727919847762
36
π
Error: 4.954675558×
-113
10