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

Posted 3 years 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 . 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. where the p's come from the following.  p = 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 4.432452533270231425156974872295170503055627479703516515309207411311257382068818424053632*10^-1911 and How about you? Answer
 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 - m] (*= 7.31698*10^-26.*) N[m - 445204143793/3114278261895 Tan] (*= 1.14633*10^-25.*) N[m - 16660240335878/21956210773773 Cos] (*= 1.55036*10^-29.*) N[9164071225/767519475537561 Sqrt - m] (*= 1.0495*10^-26.*) N[641908393603767/62104598047789277 Log - m] (*= 2.6098*10^-33.*) Answer