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What are the most unusual approximations to constants you found using MMA?

Posted 1 year ago
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I found a new surprising approximation to the MRB constant. That got me wondering what surprising approximations you could come up with using Mathematica. The MRB constant is Sum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}] .

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 2000];

Here is an algorithm that gives over 38 digits per iteration using integer coefficients of an average of less than 38 digits.

m = Sum[(x^(1/x)*p[x])/(E^(28<em>Pix) - 157329), {x, 1, Infinity}]

where the p's come from the following.

 p[0] = 1; Table[
  t[n] = m - Sum[(x^(1/x)*p[x])/(E^(28*Pi*x) - 157329), {x, 1, n}] ; 
  p[n + 1] = 
   IntegerPart[
    t[n]/((n + 1)^(1/(n + 1))/(-157329 + E^(28 (n + 1) Pi)))], {n, 0, 
   50}]; m - Sum[(x^(1/x)*p[x])/(E^(28*Pi*x) - 157329), {x, 1, 50}]
 ListLinePlot[Table[p[x], {x, 50}]] .

This gives 4.43245253327023142515697487229517050305562747970

3516515309207411311257382068818424053632*10^-1911

and

A reasonably well-behaved plot

How about you?

The following might not be unusual, as they are just results of convergents. Nonetheless, don't the irrational factors add a little intrigue?

Let m be the MRB constant :

m=0.18785964246206712024851793405427323005590309490013878617200468408947\
7231564;

     N[1013127954122/2122744284827 Cos[84230760] - m]

     (*= 7.31698*10^-26.*)


     N[m - 445204143793/3114278261895 Tan[76183051]]

     (*= 1.14633*10^-25.*)


    N[m - 16660240335878/21956210773773 Cos[1056797313]]

    (*= 1.55036*10^-29.*)

N[9164071225/767519475537561 Sqrt[247553391] - m]

     (*= 1.0495*10^-26.*)

    N[641908393603767/62104598047789277 Log[78249186] - m]

    (*= 2.6098*10^-33.*)
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