As some of you know, I'm hoping to, some day, find a closed form expression for the MRB constant.

 Here is my latest little nugget.

Let x=MRB constant.

 = log(x) with an error< .

Digits := 20; x := .1878596424620671202485; a := (1-604*x)/(28+209*x); b := log(x); abs(a-b)

8.949691*10^(-13)

 On May 19 and 20, 2012 I added the following six approximations:

Let x be the MRB constant and r be the the ratio of the sides to that of the base of Calabi's Triangle then

=x with an error of 6.2897687*10^-12.

x := .1878596424620671202485179340542732300559; r := 155138752454/10^11; evalf(x+1/16+(2*(190*r-387))/(707*r-360), 20)

-6.28976872*10^(-12)

 x+2/43= with an error < 

x := .1878596424620671202485179340542732300559; evalf(x+2/43-(70649/783309)*Pi+40176/(261103*Pi), 22)

-4.94*10^(-20)

 x= with an error<

x := .1878596424620671202485179340542732300559; evalf(x+(1/2-(-830+2*Pi+3*Pi^2)/(2*(-922-173*Pi+90*Pi^2))), 22)

4.99*10^(-20)

 x= withan error of about 

x := .1878596424620671202485179340542732300559; evalf(x+(10/21+(2*(1043*Zeta(2)-3545))/(370*Zeta(2)+4901)), 22)

2.077*10^(-19)

x+2/27= the real root of  with an error of about 

x := .1878596424620671202485179340542732300559; evalf(x+2/27-fsolve(3604068*n^3-64769, n), 22)

6.78499*10^(-17)

x+13/27= the real root of  with an error of about 

 x := .1878596424620671202485179340542732300559; evalf(x+13/27-fsolve(7778924*n^3-2332717, n), 22)

-4.26586*10^(-17)   

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