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# Can you Find a Closed form for this Number?

Posted 9 years ago
 The number involves the MRB constant. It is c = 0.316453098860696332954691288864309266518251..., I've attached over 3014991 digits of it. Here is the formula I used to compute it: -(2*x - 3) - Log[m*Cos[x*ArcCos[3]]]/Log[-1 + Sqrt[2]] /. x -> 3014991  where m is 3014991 digits of the MRB constant. I've found out that  Limit[(Sqrt[2] - 1)^(-(2*x - 3) - c)/Cos[x*ArcCos[3]], x -> Infinity]= MRB constant  .!!! As examples, it is used in the following approximation. In[131]:= (Sqrt[2] - 1)^(-7.31645309886069633295469128886430926651825135500119965205926\ 4044803278048652714707)/Cos[5 ArcCos[3]] Out[131]= \ 0.18785963830947022338298272934951014890322206494050103351553632625518\ 3736975759412 In[132]:= (Sqrt[2] - 1)^(-97.3164530988606963329546912888643092665182513550011996520592\ 64044803278048652714707)/Cos[50 ArcCos[3]] Out[132]= \ 0.18785964246206712024851793405427323005590309490013878617200468408947\ 7231564654981  , where the MRB constant is 0.18785964246206712024851793405427323005590309490013878617200468408947\ 7231564660213703296654433107496903842345856258019061231370094759226...  I see that we have the following here: , where m is the MRB constant. That is 2/(1 + Sqrt[2])^(3 - c) == m. Thus That is ((m/2)(7 + 5Sqrt[2]))^(1/c) == 1 + Sqrt[2] Attachments: