I will use this post for a list of conclusions drawn from MRB constant N and the many similar approximations that I have found. 

Let x= MRB Constant.   Each approximation is followed by a maple input so you can verify these approximations. 

The digits of accuracy (D.O.A.) might seem embellished by 1 digit because it is computed by the reciprocal of the magnitude of error. For instance, The Champernowne Constant = 0.1234567891011121314151617181920212223243..., and its approximation constructed from the MRB constant is (1/2)*(8647+1490*x)/(31589+24300*x).  0.1234567891011121314151617181920212223243-(1/2)*(8647+1490*x)/(31589+24300*x)= 9.79861545008*10^(-19). Thus, according to this rule, there are 19 digits worth of accuracy in the approximation. However, You could argue that counting the 0, only 18 digit actually match .

Shallit Constant(26406/37505 + (26568*x)/7501) to all known digits of accuracy  (26406/37505 + (26568*x)/7501)

Murata's Constant =  (4*(7124-75*x))/(10057+27*x) to 17 digits of accuracy.  (4*(7124-75*x))/(10057+27*x)

Zeta (2) =  (1/2)*(358708+823*x)/(109034+251*x) to 17 digits of accuracy.    (1/2)*(358708+823*x)/(109034+251*x)

Silver constant = (4*(58709+145*x))/(72324+181*x)  to 17 digits of accuracy.  (4*(58709+145*x))/(72324+181*x)

Silver constant =    (4/5)*(21265+2211*x)/(5352-55*x)   to 17 D.O.A.   (4/5)*(21265+2211*x)/(5352-55*x)

The domino tiling constant =   (2*(5812+189*x))/(40451-1806*x)   to 17 digits of accuracy. (2*(5812+189*x))/(40451-1806*x)

Lemniscate constant  = (12*(11123+50*x))/(50890+309*x)  to 17 digits of accuracy.   (12*(11123+50*x))/(50890+309*x)

Lemniscate constant  =   (2*(13981+2709*x))/(10819+1242*x)    to 18 digits of accuracy. (2*(13981+2709*x))/(10819+1242*x)

The 2nd Lemniscate Constant =   (8/3)*(-13556+839*x)/(-8783+576*x)   to 17 digits of accuracy.  (8/3)*(-13556+839*x)/(-8783+576*x)

Plouffe's A-constant i.e.0.15915...  =  (1/2)*(24491+8*x)/(77485-2872*x)     to 18 digits of accuracy.  (1/2)*(24491+8*x)/(77485-2872*x)

Plouffe’s gamma constant(1/380)*(32353+89*x)/(577+x) to 18 digits of accuracy.  (1/380)*(32353+89*x)/(577+x)

Heath-Brown-Moroz constant  = (73-39*x)/(49803+206*x)  to 18 digits of accuracy.   (73-39*x)/(49803+206*x)

Heath-Brown-Moroz Constant = (1/5)*(379+2*x)/(57285+1592*x) to 19 digits of accuracy.   (1/5)*(379+2*x)/(57285+1592*x)

Heath - Brown - Moroz Constant =  (2/9)*(-272+27*x)/(-45587+3030*x)    to 18 D.O.A.  (2/9)*(-272+27*x)/(-45587+3030*x)

Robbins constant-(1848/308545)*exp(1)+41856/61709-(96/61709)*x  to 18 digits of accuracy.   -(1848/308545)*exp(1)+41856/61709-(96/61709)*x

Robbins constant =  (5/583)*(12430+343*x)/(161+5*x) to 17 digits of accuracy. (5/583)*(12430+343*x)/(161+5*x)

Robbins constant =  (21/8)*(-5465+164*x)/(-21689+700*x)   to 17 digits of accuracy. (21/8)*(-5465+164*x)/(-21689+700*x)

The first Hundred-Dollar, Hundred-Digit Challenge =  (2*(-5019+4*x))/(-31069+168*x)  to 17 digits of accuracy.  (2*(-5019+4*x))/(-31069+168*x)

The 2nd Hundred-Dollar Challenge constant(2050903/1616720)*exp(1)-5/2+(1/4)*x  to 21 digits of accuracy.  (2050903/1616720)*exp(1)-5/2+(1/4)*x

The 2 nd Hundred - Dollar Challenge constant =  (4*(-6017+4010*x))/(-25363+22400*x)    to 18 D.O.A..  (4*(-6017+4010*x))/(-25363+22400*x)

The decimal expansion of solution to problem #5 in the Trefethen challenge. =1/96187 ((205030/27)*exp(1)-25-3*n ) to 19 D.O.A.   1/96187((205030/27)*exp(1)-25-3*n)

The 6th Hundred-Dollar Challenge constant =  (2651+232*x)/(43039+2568*x)   to 17 digits of accuracy.  (2651+232*x)/(43039+2568*x)

7th hundred-dollar challenge constant -(1/4)*(-12233+267*x)/(4187+72*x) to 17 digits of accuracy.  -(1/4)*(-12233+267*x)/(4187+72*x)

8th hundred-dollar challenge constant  =  (10/3)*(-1038+517*x)/(-7688+1551*x)    to 19 D.O.A.  (10/3)*(-1038+517*x)/(-7688+1551*x)

10th Hundred-Digit Challenge constant = -7*x/(3424290+12707*x)  to 19 digits of accuracy.  -7*x/(3424290+12707*x)

Probability that at least two people in a room of 23 share the same birthday(1/3100)*(1113+874*x)/(1-x) to 19 digits of accuracy.   (1/3100)*(1113+874*x)/(1-x)

Probability that at least two people in a room of 23 share the same birthday =   (2*(2879-330*x))/(10517+3135*x)   to 17 D.O.A.   (2*(2879-330*x))/(10517+3135*x)

Universal Parabolic Constant(1/10)*(318053+623*x)/(13855+27*x) to 19 digits of accuracy.   (1/10)*(318053+623*x)/(13855+27*x)



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