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MRB constant approximations using TranscendentalRecognize

By using TranscendentalRecognize, I think I came up with the most efficient approximation of MRB from given forms.

It might help in analysing the code's efficiency by checking out he attached sample MRB inver...nb .

I started with a program posted online by someone going by the name of Simon. . A similar one is found deep within in Mathworld.and at http://forums.wolfram.com/mathgroup/archive/2005/Jan/msg00254.html .

 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]

I built around it a data mining operation.like the following, where h is a list of many constants of the form x(1/x). I lost the list for h there, but it had 40 or more constants, like, h = Table[x^(1/x), {x, 2, 40}].

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity},Method -> "AlternatingSigns", WorkingPrecision -> 1000]; .

.

      d = 100; er = 20; Table[{Print[x, y], (t = 
        TranscendentalRecognize[N[(m), d], c[x, y] = h[[x ;; y]]]; 
      s = N[m - t, d]); If[s^2 < 10^-(2 d + er), Print[{t, s}]]}, {x,   1,  10}, {y, 10, 10}]

I got some, what I think are, efficient ( i.e.fewer variables with smaller constants), approximations to my MRB constant. (MRB constant is defined in Wolfram' MathWorld and is known by W[A..) These all were limited by only using the form of c^(n/x), where n and x, both<100.

In[197]:= Quiet[
 N[m - (-129858773922357615372945307143544254 - 
      332618118135196201861563173048187520*3^(1/5) + 
      360989141074787715535168098417020609*2^(2/5)*3^(1/5))/
    261711912538111957032871243762602971, 200]]

Out[197]= \
3.67832800603717470665368223835575291198522877559637247418956870295862\
1540717957587861917095449501665482864609259912010997464854189390496601\
7326568194578935410565495353817533097199630888364091944358909*10^-144

  In[197]:= Quiet[
   N[m - (-129858773922357615372945307143544254 - 
  332618118135196201861563173048187520*3^(1/5) + 
  360989141074787715535168098417020609*2^(2/5)*3^(1/5))/
  261711912538111957032871243762602971, 200]]

    Out[197]= \
    3.67832800603717470665368223835575291198522877559637247418956870295862\
  1540717957587861917095449501665482864609259912010997464854189390496601\
  7326568194578935410565495353817533097199630888364091944358909*10^-144

Here are a few more approximations that I found, which ended up giing a lot of digits more that the value I gave to my mining program: (These are limmited to a few other restrictions.)

   In[218]:= Quiet[
    N[m - (791403284963475530487868944442284292407553*StieltjesGamma[2] + 

      3364575415987830296212016870286414208213855*StieltjesGamma[4] + 

      6501952717766008515310532964426708542883509*
       Sum[8^(-n!), {n, 1, Infinity}] + 

      2814879473510959962780731825640664542716703*
       Sum[9^(-n!), {n, 1, Infinity}])/
         6717984055252083042876120552656613060137774, 300]]

Out[218]= \
-6.2242642909663704162355532151579039524967358848368194746368886689290\
751661818839021804\
  93643940426236207972707862693901673385279106988471319969698985140112\
7435709703873332`134\
  .9155634653911*^-216

\


In[219]:= Quiet[
 N[m - (-372967063067619742205241026585337180*StieltjesGamma[2] + 
            320687606599321465073123905986839263*StieltjesGamma[4] + 
            102211052827902295002288334566721956*StieltjesGamma[12] + 

      212441894526555216749911459715341854*
       Sum[8^(-n!), {n, 1, Infinity}] + 

      155185652908810652615766086341025368*
       Sum[9^(-n!), {n, 1, Infinity}])/
         284315814249073009815326355615579013, 300]]

Out[219]= \
-9.9524676215315096691725832693825358984906951576789134259705676021664\
248059747678548554\
  60610615180989951232428253214135672396501134144042507059888478602960\
236138224312425`134.\
  12203729059152*^-217

Null

In[225]:= Quiet[
 N[m - (7104443136721728575905321650535196463313633*3^(2/9) - 

      2180718111811910815323291747705043372745259*StieltjesGamma[9] - 

      2896588673082600778149973623311628930807255*
       Sum[12^(-n!), {n, 1, Infinity}] - 
            6015801821759926483557786721397041400269244*Zeta[3])/
         8390253020596747014154150758555797074475505, 300]]

Out[225]= \
-3.3222593389038579058691628775383126316686843433100927697761938470770\
423450105378910571\
  33932112092296214244839309765166771670398490360706592080820770773400\
60779447336714`132.8\
  8558801750568*^-217

Here is another one , I think is good, Those of you that might know how to judge such appox. ,how am I doing? Because

TranscendentalRecognize[],, I think, was made to give the optimized approximation I will attach a few of my number mining notebooks.with the most precise approx.the precision in10^-x will be named that x.

Attachments:
POSTED BY: Marvin Ray Burns A.G.S. (cum laude)
4 Replies
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Attachments:
POSTED BY: Marvin Ray Burns A.G.S. (cum laude)

Below and attached are two huge approximations of the MRB constant using only integers and the form Sum[x^(-n!), {n, 2, Infinity}],, for x from 2 to 101..

This first one gives around 3000 accurate decimals.

And the one below it gives around 4000 accurate decimals!

In[7]:= 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]

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 4000]; 



s = Table[Sum[x^-(n!), {n, 2, Infinity}], {x, 2, 102}]





In[50]:= TranscendentalRecognize[N[m, 3000], s[[2 ;; 101]]]






Null

N[m - (-1617601210758780475076147232954091*
      Sum[3^(-n!), {n, 2, Infinity}] + 

     1260730972958628376379103266584756*
      Sum[4^(-n!), {n, 2, Infinity}] + 

     168255942381333226880232447282467*
      Sum[5^(-n!), {n, 2, Infinity}] + 
     526758508745229051826843765249190*
      Sum[6^(-n!), {n, 2, Infinity}] - 
     2670042164293488356909920338891358*
      Sum[7^(-n!), {n, 2, Infinity}] + 

     3829272605182291498586693806942722*
      Sum[8^(-n!), {n, 2, Infinity}] + 

     859415018288984049908165889504129*
      Sum[9^(-n!), {n, 2, Infinity}] + 
     3980672361148310552936792479918956*
      Sum[10^(-n!), {n, 2, Infinity}] + 
     2949125963814399232221423020582609*
      Sum[11^(-n!), {n, 2, Infinity}] - 

     1704497602955159964193025017671421*
      Sum[12^(-n!), {n, 2, Infinity}] - 

     1158615646978266430347663044541968*
      Sum[13^(-n!), {n, 2, Infinity}] + 

     2940566178482760393686097895245937*
      Sum[14^(-n!), {n, 2, Infinity}] + 

     3537625077232750001025112741131143*
      Sum[15^(-n!), {n, 2, Infinity}] + 

     1628630162143702095790590207317237*
      Sum[16^(-n!), {n, 2, Infinity}] + 

     2075441527976471396863264573730348*
      Sum[17^(-n!), {n, 2, Infinity}] - 

     2264173752934939591163612937875735*
      Sum[18^(-n!), {n, 2, Infinity}] - 

     2522296388129947914734380874599382*
      Sum[19^(-n!), {n, 2, Infinity}] + 

     2320118429832380290630535582086539*
      Sum[20^(-n!), {n, 2, Infinity}] - 

     2463815400152015299479476128150570*
      Sum[21^(-n!), {n, 2, Infinity}] + 

     2638986590182329976236035098810060*
      Sum[22^(-n!), {n, 2, Infinity}] + 

     3251654159711357468899799174679152*
      Sum[23^(-n!), {n, 2, Infinity}] + 

     93915231319017548602476626871287*
      Sum[24^(-n!), {n, 2, Infinity}] + 
     193808727512029207895977355455647*
      Sum[25^(-n!), {n, 2, Infinity}] + 
     3607117201144790057874151852014759*
      Sum[26^(-n!), {n, 2, Infinity}] + 

     3262007418561507649847272272032955*
      Sum[27^(-n!), {n, 2, Infinity}] + 

     2480402689383408099208714687051739*
      Sum[28^(-n!), {n, 2, Infinity}] + 

     1308509867990398861324717427268301*
      Sum[29^(-n!), {n, 2, Infinity}] + 

     1311639103727367378793170201050131*
      Sum[30^(-n!), {n, 2, Infinity}] - 

     1648727214133032285519999314329705*
      Sum[31^(-n!), {n, 2, Infinity}] + 

     3895395322625799512405275036584486*
      Sum[32^(-n!), {n, 2, Infinity}] - 

     4228124835647564954643945470573976*
      Sum[33^(-n!), {n, 2, Infinity}] + 

     1724355876667841445871244995563479*
      Sum[34^(-n!), {n, 2, Infinity}] - 

     821120993451512960591013772378796*
      Sum[35^(-n!), {n, 2, Infinity}] - 

     2219010432574623362013336951784027*
      Sum[36^(-n!), {n, 2, Infinity}] + 

     955107235645995663705816457381081*
      Sum[37^(-n!), {n, 2, Infinity}] + 

     4977626512233618799592076868098467*
      Sum[38^(-n!), {n, 2, Infinity}] + 

     479750350430292719426344888964982*
      Sum[39^(-n!), {n, 2, Infinity}] - 

     1009428065023075859234682582260676*
      Sum[40^(-n!), {n, 2, Infinity}] - 

     453768351398454243808381569101699*
      Sum[41^(-n!), {n, 2, Infinity}] - 

     804331952205012340117978795714571*
      Sum[42^(-n!), {n, 2, Infinity}] + 

     2112537773325380956038299597223034*
      Sum[43^(-n!), {n, 2, Infinity}] + 

     1563909600882194431077161802113231*
      Sum[44^(-n!), {n, 2, Infinity}] + 

     3281512201875089338818140791672361*
      Sum[45^(-n!), {n, 2, Infinity}] - 

     841693145563449440980152624889040*
      Sum[46^(-n!), {n, 2, Infinity}] - 

     1180596415104015890930529257781618*
      Sum[47^(-n!), {n, 2, Infinity}] + 

     256894002482080676102804595099554*
      Sum[48^(-n!), {n, 2, Infinity}] + 

     3059743069488164456875317821197684*
      Sum[49^(-n!), {n, 2, Infinity}] + 

     618378447020184993694643494855854*
      Sum[50^(-n!), {n, 2, Infinity}] + 

     1207849584860413404886567472602012*
      Sum[51^(-n!), {n, 2, Infinity}] + 

     3269594033521468442241723242546780*
      Sum[52^(-n!), {n, 2, Infinity}] - 

     519110770734317313929892528340481*
      Sum[53^(-n!), {n, 2, Infinity}] + 

     5337933979867080683965490693782692*
      Sum[54^(-n!), {n, 2, Infinity}] - 

     845305587078512215850327456666298*
      Sum[55^(-n!), {n, 2, Infinity}] + 

     3638411700443380962703568627301563*
      Sum[56^(-n!), {n, 2, Infinity}] + 

     57104890901297559220617943398070*
      Sum[57^(-n!), {n, 2, Infinity}] - 
     349193535926582284655062345576058*
      Sum[58^(-n!), {n, 2, Infinity}] - 
     2574102460519834845887788602861507*
      Sum[59^(-n!), {n, 2, Infinity}] + 

     626918354620070457316744226035168*
      Sum[60^(-n!), {n, 2, Infinity}] + 

     925910782814132818900483793749726*
      Sum[61^(-n!), {n, 2, Infinity}] + 

     2659043142343524075790297170462132*
      Sum[62^(-n!), {n, 2, Infinity}] - 

     3776624195115615476424023462853132*
      Sum[63^(-n!), {n, 2, Infinity}] + 

     289252819950910792633107902104858*
      Sum[64^(-n!), {n, 2, Infinity}] - 

     4335832676811110969381326369850434*
      Sum[65^(-n!), {n, 2, Infinity}] - 

     1049666000572944258193455109018929*
      Sum[66^(-n!), {n, 2, Infinity}] - 

     2060181142655155567883399336847960*
      Sum[67^(-n!), {n, 2, Infinity}] - 

     1802686787596758226171368511609941*
      Sum[68^(-n!), {n, 2, Infinity}] + 

     560149266456680122048780702178403*
      Sum[69^(-n!), {n, 2, Infinity}] + 

     45326892699113686458133865345872*
      Sum[70^(-n!), {n, 2, Infinity}] - 
     2189834207054837650558549556270607*
      Sum[71^(-n!), {n, 2, Infinity}] - 
     200007960908326787314318800286238*
      Sum[72^(-n!), {n, 2, Infinity}] + 

     1906908709896907749806302689843055*
      Sum[73^(-n!), {n, 2, Infinity}] - 

     2413831603161356405527598276237701*
      Sum[74^(-n!), {n, 2, Infinity}] + 

     349851432963186321571769787692286*
      Sum[75^(-n!), {n, 2, Infinity}] - 

     1409077148223763233830923073076654*
      Sum[76^(-n!), {n, 2, Infinity}] - 

     3258926296143830773420165267895655*
      Sum[77^(-n!), {n, 2, Infinity}] - 

     2788017156113146729276123053829960*
      Sum[78^(-n!), {n, 2, Infinity}] + 

     2692847787140868892915206384398765*
      Sum[79^(-n!), {n, 2, Infinity}] - 

     567067016846573451010485778130831*
      Sum[80^(-n!), {n, 2, Infinity}] - 

     1905831513124390141890451640600611*
      Sum[81^(-n!), {n, 2, Infinity}] - 

     5136597947331669055009371904973487*
      Sum[82^(-n!), {n, 2, Infinity}] + 

     912278368494920565761271769690129*
      Sum[83^(-n!), {n, 2, Infinity}] - 

     2580979677015581716819042733783251*
      Sum[84^(-n!), {n, 2, Infinity}] + 

     1884158460408959578540033438166537*
      Sum[85^(-n!), {n, 2, Infinity}] + 

     2996809848467580062948210521300363*
      Sum[86^(-n!), {n, 2, Infinity}] + 

     2547118062655141771555127378895892*
      Sum[87^(-n!), {n, 2, Infinity}] + 

     122733378092474394591379091380222*
      Sum[88^(-n!), {n, 2, Infinity}] + 

     1609043837874589833719447903600838*
      Sum[89^(-n!), {n, 2, Infinity}] + 

     706449525697658734698912642263703*
      Sum[90^(-n!), {n, 2, Infinity}] + 

     331198650888790532168855884736297*
      Sum[91^(-n!), {n, 2, Infinity}] + 

     616901082829008843727297647565846*
      Sum[92^(-n!), {n, 2, Infinity}] + 

     2407854441179185932069643380268222*
      Sum[93^(-n!), {n, 2, Infinity}] - 

     4494561604681894784950422947849268*
      Sum[94^(-n!), {n, 2, Infinity}] + 

     1259577304594505752616397860280623*
      Sum[95^(-n!), {n, 2, Infinity}] - 

     2564935607306630357867809604407419*
      Sum[96^(-n!), {n, 2, Infinity}])/
   230024844450836751070309032939613, 
   3000]

giving

  Out[49]= 1 36025200332815066380729833521872476571420563018384745339137125938852363572682438`47.\
36565286514424*^-3002

Here is the 4,000 approximation:

N[m - (-296516481636968137503982279733117020027871*
      Sum[3^(-n!), {n, 2, Infinity}] + 
     325661615276655660175959176674077806917110*
            Sum[4^(-n!), {n, 2, Infinity}] + 
     642182094925939713175056356584465598954027*
      Sum[5^(-n!), {n, 2, Infinity}] + 

     601175274312643299513368500777196611943043*
      Sum[6^(-n!), {n, 2, Infinity}] + 
     461071564223118140831888034215659217102634*
            Sum[7^(-n!), {n, 2, Infinity}] - 
     46056184702533685913819676113042970084596*
      Sum[8^(-n!), {n, 2, Infinity}] + 

     279339004802534787952844028108115284244029*
      Sum[9^(-n!), {n, 2, Infinity}] - 
     299266395268565444689230937067754075583779*
            Sum[10^(-n!), {n, 2, Infinity}] - 
     465234809034158007809899570315803804300290*
      Sum[11^(-n!), {n, 2, Infinity}] + 

     285887132280583324375825088698929314025198*
      Sum[12^(-n!), {n, 2, Infinity}] - 
     1455153254962389166044713884804999761815895*
            Sum[13^(-n!), {n, 2, Infinity}] - 
     433114104837923880113595542995205621923404*
      Sum[14^(-n!), {n, 2, Infinity}] + 

     1160117914615393859326075479663370802329893*
      Sum[15^(-n!), {n, 2, Infinity}] - 
     847974925978145426909219295179634996686481*
            Sum[16^(-n!), {n, 2, Infinity}] - 
     478039072552888313172646196547768919885589*
      Sum[17^(-n!), {n, 2, Infinity}] - 

     1207220432365216840785401069506724372409135*
      Sum[18^(-n!), {n, 2, Infinity}] - 
     565442430159872587593563683618559838970756*
            Sum[19^(-n!), {n, 2, Infinity}] + 
     281004860950920620115935860078699244770248*
      Sum[20^(-n!), {n, 2, Infinity}] + 

     38542239404878995475716101692658191802366*
      Sum[21^(-n!), {n, 2, Infinity}] + 
     1442550945203909873532394738550482603093249*
            Sum[22^(-n!), {n, 2, Infinity}] - 
     1070315093526954824158169515996581349047557*
      Sum[23^(-n!), {n, 2, Infinity}] - 

     1196569785451215366214358216221843708950368*
      Sum[24^(-n!), {n, 2, Infinity}] - 
     1504057272276809031352388906961647123416340*
            Sum[25^(-n!), {n, 2, Infinity}] + 
     475291034780375689238862608191953323145543*
      Sum[26^(-n!), {n, 2, Infinity}] - 

     524027840995878115513925912794868447867202*
      Sum[27^(-n!), {n, 2, Infinity}] - 
     1620441974573474591410837709104326096167725*
            Sum[28^(-n!), {n, 2, Infinity}] - 
     414214539885862686599974983033151068422599*
      Sum[29^(-n!), {n, 2, Infinity}] + 

     559985517658515157527015054675982392369751*
      Sum[30^(-n!), {n, 2, Infinity}] + 
     487621074248268136802789768671137109262588*
            Sum[31^(-n!), {n, 2, Infinity}] + 
     674666619828450995162006462639617999407097*
      Sum[32^(-n!), {n, 2, Infinity}] - 

     106618498602952986491811414739566870171557*
      Sum[33^(-n!), {n, 2, Infinity}] + 
     1168555124211830147037898399967040382528817*
            Sum[34^(-n!), {n, 2, Infinity}] + 
     85390599906882793536181196132857692177667*
      Sum[35^(-n!), {n, 2, Infinity}] + 

     569708755874101365117562354940363507455858*
      Sum[36^(-n!), {n, 2, Infinity}] - 
     435805753404222631106858418183069441499085*
            Sum[37^(-n!), {n, 2, Infinity}] - 
     100632304065941203378884812209693888341737*
      Sum[38^(-n!), {n, 2, Infinity}] + 

     175980026251533802235364343828740109328330*
      Sum[39^(-n!), {n, 2, Infinity}] - 
     203093029693440514891257626039459460579530*
            Sum[40^(-n!), {n, 2, Infinity}] - 
     527109494975226274138625409584388590824167*
      Sum[41^(-n!), {n, 2, Infinity}] - 

     1441919975136263248521894296063643037329957*
      Sum[42^(-n!), {n, 2, Infinity}] + 
     302034343438868384200878876044014773620801*
            Sum[43^(-n!), {n, 2, Infinity}] + 
     583248854885837765648122572543150789918959*
      Sum[44^(-n!), {n, 2, Infinity}] + 

     702270194902490380525021700597036955614413*
      Sum[45^(-n!), {n, 2, Infinity}] - 
     500436526244688040036598186854976926921324*
            Sum[46^(-n!), {n, 2, Infinity}] + 
     81489006025013640398934425492602897258138*
      Sum[47^(-n!), {n, 2, Infinity}] - 

     761473920288312286531262650480401117313772*
      Sum[48^(-n!), {n, 2, Infinity}] - 
     1034990444893569446002055543630756560017555*
            Sum[49^(-n!), {n, 2, Infinity}] - 
     754253538292494367301525764238807361811031*
      Sum[50^(-n!), {n, 2, Infinity}] - 

     1132795850299834123210924644846840733101239*
      Sum[51^(-n!), {n, 2, Infinity}] + 
     2387010927143877767508686859158595028391373*
            Sum[52^(-n!), {n, 2, Infinity}] - 
     1089822591277202578314471461685208924013469*
      Sum[53^(-n!), {n, 2, Infinity}] - 

     81001683977437981798666572764812171960730*
      Sum[54^(-n!), {n, 2, Infinity}] - 
     582248537457995534060569443329206686787816*
            Sum[55^(-n!), {n, 2, Infinity}] + 
     168776064302334723300910242823139170872068*
      Sum[56^(-n!), {n, 2, Infinity}] - 

     424077533292026559795307574966795337212312*
      Sum[57^(-n!), {n, 2, Infinity}] + 
     1109947769465243875535508831964972433450393*
            Sum[58^(-n!), {n, 2, Infinity}] + 
     612691804838323729773604875710797428704379*
      Sum[59^(-n!), {n, 2, Infinity}] - 

     846613643602017244023404751070377238390310*
      Sum[60^(-n!), {n, 2, Infinity}] - 
     637234398046077593684936948711478827364827*
            Sum[61^(-n!), {n, 2, Infinity}] - 
     595473419173144376910893545292289278055168*
      Sum[62^(-n!), {n, 2, Infinity}] - 

     798382236515911107795355064641213265865957*
      Sum[63^(-n!), {n, 2, Infinity}] - 
     1475710630751426353616979652407251998437441*
            Sum[64^(-n!), {n, 2, Infinity}] - 
     349266086857445890482716730556079686616657*
      Sum[65^(-n!), {n, 2, Infinity}] + 

     576297702314586207635300890940428731650513*
      Sum[66^(-n!), {n, 2, Infinity}] + 
     64424778631246518288433825435442342473108*
            Sum[67^(-n!), {n, 2, Infinity}] + 
     768355660157040286478201814231161522585435*
      Sum[68^(-n!), {n, 2, Infinity}] + 

     293710790604950219816671224005902252977446*
      Sum[69^(-n!), {n, 2, Infinity}] - 
     559797054427464244707254151567534633845493*
            Sum[70^(-n!), {n, 2, Infinity}] + 
     80950405776998237843876379200957275189067*
      Sum[71^(-n!), {n, 2, Infinity}] + 

     1670729414137120803674093465162507318808519*
      Sum[72^(-n!), {n, 2, Infinity}] + 
     775685504163875455988543406432695543816217*
            Sum[73^(-n!), {n, 2, Infinity}] + 
     1333091995709003174575758585461773893418308*
      Sum[74^(-n!), {n, 2, Infinity}] - 

     155872950652287384316462697258741573600564*
      Sum[75^(-n!), {n, 2, Infinity}] + 
     1012411988025439131263676384839491452505632*
            Sum[76^(-n!), {n, 2, Infinity}] + 
     1435991513389482138446348675629225179237248*
      Sum[77^(-n!), {n, 2, Infinity}] - 

     163117194140768456768697927667437080083367*
      Sum[78^(-n!), {n, 2, Infinity}] - 
     1894571430791935332300955306245180675566119*
            Sum[79^(-n!), {n, 2, Infinity}] - 
     88650069624075571861901184013020969336140*
      Sum[80^(-n!), {n, 2, Infinity}] - 

     2443195669827387130450675747324794999468278*
      Sum[81^(-n!), {n, 2, Infinity}] - 
     445403791149756764405424065112370443906832*
            Sum[82^(-n!), {n, 2, Infinity}] + 
     1011876040895889660018147477114342871487124*
      Sum[83^(-n!), {n, 2, Infinity}] - 

     841225667122861617814628363282657839089313*
      Sum[84^(-n!), {n, 2, Infinity}] + 
     781153505932427407408615277783615012304737*
            Sum[85^(-n!), {n, 2, Infinity}] - 
     1512669741220256920604392332262610689126748*
      Sum[86^(-n!), {n, 2, Infinity}] - 

     116806211318565735630179723946845979649289*
      Sum[87^(-n!), {n, 2, Infinity}] + 
     429130091032045229595229240522857578374793*
            Sum[88^(-n!), {n, 2, Infinity}] + 
     506765034962670771473975504854800595831172*
      Sum[89^(-n!), {n, 2, Infinity}] - 

     879274842685359858085342642938644291494265*
      Sum[90^(-n!), {n, 2, Infinity}] - 
     74882381833281682041429986677690145192329*
            Sum[91^(-n!), {n, 2, Infinity}] - 
     1481962754796105396642750584065054290318576*
      Sum[92^(-n!), {n, 2, Infinity}] + 

     371582750848581648295884969326092328244938*
      Sum[93^(-n!), {n, 2, Infinity}] - 
     548940842526326295585707606326385101045388*
            Sum[94^(-n!), {n, 2, Infinity}] - 
     214364159079515020789428860678593442975390*
      Sum[95^(-n!), {n, 2, Infinity}] - 

     792746436246894058263311867715612724830428*
      Sum[96^(-n!), {n, 2, Infinity}] + 
     98779136320327876056553220251060452284968*
            Sum[97^(-n!), {n, 2, Infinity}] + 
     420817031258715595750978255792960491285033*
      Sum[98^(-n!), {n, 2, Infinity}] + 

     133685347739575372427537403082058930459109*
      Sum[99^(-n!), {n, 2, Infinity}] + 
     624039311167296059305190395710287624563274*
            Sum[100^(-n!), {n, 2, Infinity}] - 
     1033617019679927139078081654715284819642299*
      Sum[101^(-n!), {n, 2, Infinity}] - 

     684584444208170176786209670810376971046901*
      Sum[102^(-n!), {n, 2, Infinity}])/
   85378145960320201007017729300346246766531, 5000]
Attachments:
POSTED BY: Marvin Ray Burns A.G.S. (cum laude)

Here are my latest approximations to the MRB constant.They are the best looking one out of a few thousand. These approximate 1000 digits of the constant, but give the most beyond, 1000 digits, accuracy.

N[m - (510017535631180687460856957450572305326540096915661936978654660685564647512350782555553614907470983387825537836511551063206613482712467827893\
01468559609485014131037773093095425665777333045322543428553522*(-1 + Sqrt[2]) - 96068150340234077374726370910347925333938216211589364597933496589097\
075853239867010973807246924811862210135463028988657144609634882312183141641523864425497419159770197204815060852100498555727184938689116*
(-1 + 2^(1/18)*3^(1/9)) + 23333703534904618706173261841895326308700547835506225964185595796344451478775469597207817863820838379134974452053729\
3369333105298477043109281116710787527167404406291408274367957775851669383906747626354830*Sum[4^(-n!), {n, 2, Infinity}] + 
40313385556940898081252974448093819151002029893520177263320252381062732980131887242120530425911918252062181272894937220571111439759199788063701\
4439799379786658216842616733570082844959130397748389631088288*Sum[14^(-n!), {n, 2, Infinity}])/11226020246276929216469224351763847870811895611134682\
9809426965838194867887854374179420796093728124712438534958688705043058060873338334003999743465958730946783909428577286552639810756878788810016950541\
959, 2000]

. It gives

9.1978955072905472962...2*10^-1018 .

N[m - (404355126688171915451034416312230604919707970452383451853925931\
40406938951988728889681015103623256198012637149747345520129713\
       49689454162107757602250020811957699990816546*(1 - 7^(1/7)) + 
     33051480591839495225039933253849294611927448814171710704107740940\
734665\
       040457829733431797688126677213799052071795406260439448132384992\
74294322897183697379311258530999832*(-1 + 14^(1/14)) - 

     24189499316961197596532386979514491525599470961143368177569250102\
82955146179847110763589857004031035173663050083808012723899090\
       473565355251693036218605461019069457446148*(1 - 17^(1/17)) + 
     82468081425975915242688262123067077311822879410444962143373037763\
622634\
       606906917316082600654001559505105511211964587221098020620728591\
82678249467359584256403093648400278*Sum[2^(-n!), {n, 2, Infinity}] - 

     23196615765931373730391709909338745951421286383132885907911476173\
69723860751338004736792098716384442993092400628717132353654831\
       279801942857974608507685271164128259600611*
      Sum[20^(-n!), {n, 2, Infinity}])/
   10716692834803667275169650412872773031712463947868978911\
    401491029190218128259539722750075189375635799016814648578340611058\
     418703882776034702173059087881414809009655698842, 2000]

gives

1.29057693406447674817351694260292503364903562408778360761951285430083\
...
562007060546444086256424992932423084037209691675528920*10^-1017

and

N[m - (479726275966738930950716668460453456135609546153820124961462648\
4512655106535889637667232127769509716461740872574122078642387776516851\
09388282\
       12443377294736414176483302924133740398634813902217928087638057*
      Sum[4^(-n!), {n, 2, Infinity}] + 
     5838998640136470517323044439381905625083886232218587\
       669749531087797743382913252545417668577092986554098832543936290\
9113467768066621051147331513413336416581833310908902378720986774747056\
064016042598151\
       125*Sum[7^(-n!), {n, 2, Infinity}] - 
     63218949949966052258049515603589770732070427320021246973471652245\
5144788248490957660650633781863001770791932590\
       071033829885733702925886237886192728969625214564447134735540638\
5925595638579007900489820209*Sum[15^(-n!), {n, 2, Infinity}] + 

     15198929012735579673282610542007594760375937317388650950231682497\
1899195175150323553974961146574023964586942011016795308214767786746900\
16895505\
       392739086196475640858326345017228496199996878438805028527771*
      Sum[17^(-n!), {n, 2, Infinity}])/
   224988407960821873723472823677123883665729840705233851\
    269518866426796111745109675950172197114835253338797358527687132274\
3557882097017477522481690748025398465304015951108345956023807857443692\
657239661543\
    2, 2000]

gives

2.57879929491783173010969833912581455697003587799754559692884173462338\
---
578974308935437637235778544541222839048731919691392341242288831*10^-\
1017

Also 

N[m - (676553166992866963815372906222646303607573324645363021255484344\
5847651711736285938683235621024058639551951835735298345223727168642453\
37735108\
       73613949894745859102893275100501442382626915575445148788736214*\
(1 - 11^(1/11)) - 
     14619701004192034806300676602885461655161865631216564768365650294\
65\
       076216299072104991168738127984191377243798233621863800457100013\
0509345301162175372820032729558482082979094524738063563853873225007240\
4928*
            (1 - 19^(1/19)) + 
     68539239911237007645329097139664653912687801952352131936740047785\
40357555041564396019643409050283620285059876883679335919953\
       977606127636155870844440362109900051767228224136857904246430358\
8400719638395372*Sum[13^(-n!), {n, 2, Infinity}] + 

     51631709973922849174083847516931964611996582831207246415493453517\
1360384966066875043647023902955518731996597302090507331070815563491676\
50911951\
       564015771281862255807366068747634361550836780876919887207074*
      Sum[16^(-n!), {n, 2, Infinity}])/
   459606656778282203777615873161355445177495913644534440\
    236428009020039867679836025106859862935288356868529917262162124684\
8655339707929850088398876890644711371164045251650873684631194412373318\
788116834130\
    5, 2000]

gives

-3.6102997014559870502180584574338386601100194544114033623209409990651\
....
4013259366951605041002710996897846676570391136242678653046064605*10^-\
1017



N[m - (-14059768861495021278710761814432661641419385394995410885384053\
4321718939400795528466474229654128294482783893218374581982359790435922\
70347287\
        850912019400773283925905864775*(1 - 13^(1/13)) + 
     74819870702135710155830108496241896819204806952794126321695372878\
4421980599403317506355275210308751\
       837169775662594125358889924021938820605278128626037787819772197\
2689543*(-1 + 14^(1/14)) + 
     1026745578533744923810544980109604285225991455455168848218\
       851940485590090158684730723388336061515548637248082857303503426\
8011245942869978406865653859126448947036699880250*
      Sum[5^(-n!), {n, 2, Infinity}] - 

     23919308882782978080426273494605406622652109218599068941380171181\
0873283701722728640061108639572805924891231810861486595861053426099988\
74913830\
       397311908108691461538843588*Sum[11^(-n!), {n, 2, Infinity}] + 
     33490640981554250803555955384317846792767847299010359091716296701\
777133181948335535665\
       363570732763918461558209511660291118481482138653844052627227766\
575260126610639969338*Sum[16^(-n!), {n, 2, Infinity}])/
   264197139971924290274318703857\
    045641213393263172333862948269611997449923895383664022896461113170\
0352455914156203328519044605284050440986586394175399184484438567395205\
6001, 2000]

gives

2.95037252497728220301535655036996123361794994342988068078317201753568\
5315152510783969290388149141412681616174537035046372757542091836244471\
0147825110223321183122124086148527098121970008184357909548197*10^-\
1017}
POSTED BY: Marvin Ray Burns A.G.S. (cum laude)

I've found too many interesting forms for many digits of the MRB constant.I looked at a few, just as good, for other constants too.

Here is one that is extra special to me-- I don't know -- some would just see that I added a bunch of random numbers and got an answer..

I did a little studying about the TranscendentalRecognize function. With the restrictions given it was supposed to give the "simplest" solution for the given number digits.

z =  {\[Pi], E, GoldenRatio, Khinchin, EulerGamma, GoldenRatio, Catalan, \
Sqrt[2], 3^(1/3), Sqrt[2], 5^(1/5), 6^(1/6), 7^(1/7)}

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]

 For[y = 3, y < 11,
 Print[y];
 p = TranscendentalRecognize[N[m, 20 y], RandomChoice[z, 4]*y]; 
 Print[p];
Print[e = N[p - m, 64]]; y++]

Which factored by my observation is

enter image description here

Which equals,

9.67609665331181132 ... 05184839719402143746`300.*^-201

The result:::

10539265076340778455746831348074621/1147054894409218164005683918321030
+(1315939211433987901830906252006812 Sqrt[2])   /    114705489440921816400568391832103
+(2238154781652703157536445225769983 Sqrt[5])   /    229410978881843632801136783664206
+(464968409472439567819541453580715 6^(1/6))   /    114705489440921816400568391832103
-(1917024616230569159116802658552184 Pi )   /    114705489440921816400568391832103

I do realize, however, even if this is the simplest approximation,to that many digits, of that particular form, there is no chance of being the simplest form to use!

P.S.

I got a little exotic and found a nice closed form approximation to m^m^m^......

mr=(3266780045684960162847550288713990225631168+
1110218827497735378447017478768998194987712 Sqrt[2]+
5397133631025017706929697830832340263986880 3^(1/3)-
6209268135253131845504446727054527890012160 5^(1/5)-
3722403313326121192182250135829290878917440 6^(1/6)+
10028893256036615933248156928358431064871936 7^(1/7)-
4511138177384336628384814819384709016906304 E)/
35171415528443234478484447899288103133690

To check it I will use a tetration law, I don't know the name, But you can find it in MathWorld on the tetration page.

 N[m - mr^(1/mr), 400]

Out[65]= 1.\
8227207233303348653454461326515515800093235125329919778950382492329113\
4594869822885566472851208477062791211814387757010664442953073549308007\
6927632430241699590693763805649694595954601441174700087168936922446384\
3118477884367149009785953875366770689466302668084408526394897438402179\
4398904288853785020951207815180537234335182744342204474468543109558551\
6156647729629573414184056008550101815194046180953*10^-325

By the following input I GOT A NEARLY 1,000 DIGIT APPROX

In[74]:= z = Join[{I, \[Pi], E, Sqrt[2]}, Table[x^(1/x), {x, 1, 15}]]

Out[74]= {I, \[Pi], E, Sqrt[2], 1, Sqrt[2], 3^(1/3), Sqrt[2], 5^(
 1/5), 6^(1/6), 7^(1/7), 2^(3/8), 3^(2/9), 10^(1/10), 11^(1/11), 
 2^(1/6) 3^(1/12), 13^(1/13), 14^(1/14), 15^(1/15)}

TranscendentalRecognize[n_, basis_] := 
 Module[{c, d, digs, e, id, lat, powerten, r, s, vals}, {d, e} = 
   RealDigits[n];
  s = Sign[n];
  c = FromDigits[d];
  powerten = 10^(Length[d] - e);
  digs = (RealDigits[N[#1, -e + Length[d] + 5]] &) /@ basis;
  r = (FromDigits[Take[First[#1], -e + Last[#1] + Length[d]]] &) /@ 
    digs;
  Tlat = Transpose[
    Append[IdentityMatrix[Length[basis] + 2], 
     Flatten[{powerten, r, c}]]];
  vals = Take[First[LatticeReduce[lat]], Length[basis] + 2];
  Expand[-((s (Take[vals, {2, -2}].basis + First[vals]))/Last[vals])]]

In[75]:= 
For[y = 110, y < 2200, Print[y];
 p = TranscendentalRecognize[N[m, 5 y], RandomChoice[z, 7] y];
 Print[p];
 Print[e = N[p - m, 64]]; y++];

     m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, 
     Method -> "AlternatingSigns", WorkingPrecision -> 1000];






In[87]:= Block[{ $MaxExtraPrecision = 2000}, N[m - 
   (732342421548474675180526599895409684264644327151934824164674474461\
7922491658539305019493955220461660037676829012116678161973717 + 

      4072149642232149576053764120676969872061896542320267666183500991\
02049588747083043017470783162958681270629759687955222181909461330*2^(\
3/8) + 
      7681034247847453546335061180942226612630628658675512251591975879\
27899297132477996528019803100765350880729688992218863454548508878*2^(\
1/6)* 3^(1/12) - 
        34045134946025441841011260331526924008281141115692566262609055\
8152409346539144337134261814844546786394542003505888957271010500854*3^\
(2/9) - 
       609926526794066808991516705993794777295179002582017985851779358\
782777912145100024260322124118925720144824696881973453747630696704*7^(\
1/7) + 
        11372239599441213960858419663818160615356886708217344335954638\
9846875907225388509126032751322453242940643260453219863039395130962*\
13^(1/13) - 
       139334155741910768876027728909033188772221814812961177310973301\
700033416896043657269919708071396437672738505935341191663898499044*E)/
    196686824825458435614269629594068012795109891362859180654527231644\
718522850534710213275151197132395131009154363507115493359209, 1000]]

Out[87]= 3.\ 0284237122944438589618739933075298246671309372966698310863473780943006\ 37126254014208777456974745659*10^-997
POSTED BY: Marvin Ray Burns A.G.S. (cum laude)
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