Dottie Number

Author

Eric W. Weisstein
July 15, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/DottieNumber.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DottieNumber.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Value

Plot[{Cos[x],x},{x,-5,5}]

Closed forms

Root[{# - Cos[#] &, 0.739085}]

Root[{#-Cos[#]&,0.739085}]

Root[{(SphericalBesselY[0, #])+1&,0.739085}]

SphericalBesselY[0,x]//FunctionExpand

Solve[-1==SphericalBesselY[0,x]&&0<x<1,x]

Root[{SphericalBesselY[0,#]+1&,0.739085}]

(Sqrt[1-(((2 (InverseBetaRegularized[(1/2), (1/2), (3/2)])-1))^2)])

DottieConstant2=Sqrt[1-(2 InverseBetaRegularized[1/2,1/2,3/2]-1)^2]

N[DottieConstant2,20]

DottieConstant2//TraditionalForm

FullSimplify[DottieConstant2]

FullSimplify[DottieConstant-DottieConstant2]

FunctionExpand[DottieConstant-DottieConstant2]

Almost Integers

((160/π)^(1/13)) r==0.99999967661470791095

N[DottieConstant((((160/π)))^(1/13)),20]

(e^((e^(π/3))-(Gamma[r])))==5.0000000170647544117

K. Hammond

N[E^(E^(π/3)-Gamma[DottieConstant]),20]

BBP-Type Formulas

<<Utilities`Simplify`

Search

Monitor[Do[If[(b=FindBasis[DottieConstant,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-3^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-3^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-2^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-2^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,1}],{-1,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{-1,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{1,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{1,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{2^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{2^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{2^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{3^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{3^k,n,p}];//Timing

Monitor[Do[If[(b=FindBasis[DottieConstant,3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{3^k,n,p}];//Timing