Soldner's Constant

Author

Eric W. Weisstein
September 15, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Numerical Value

FindRoot

FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->20,AccuracyGoal->∞]

FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->100,AccuracyGoal->∞]//Timing

x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->150,AccuracyGoal->∞]//Timing

x/.FindRoot[LogIntegral[x],{x,14/10},WorkingPrecision->200,PrecisionGoal->120]//Timing

v5.0.1

{(SoldnerMu=x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->2 10^4])//Short//Timing,
LogIntegral[SoldnerMu]//Timing}

LogIntegral[SoldnerMu]//Timing

{(SoldnerMu=x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->5 10^4])//Short//Timing,
LogIntegral[SoldnerMu]//Timing}

{(SoldnerMu=x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->10^5])//Short//Timing,
LogIntegral[SoldnerMu]//Timing}

{(SoldnerMu=x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->5 10^5])//Short//Timing,
LogIntegral[SoldnerMu]//Timing}

...ran overnight...

v5.1

(SoldnerMu=x/.FindRoot[LogIntegral[x],{x,14/10},WorkingPrecision->1.7 10^4])//Short//Timing

(SoldnerMu=x/.FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->1.7 10^4])//Short//Timing

Transcendental Root

V6.1

μ=Reduce[LogIntegral[x]==0&&0<x<2,x,Reals][[2]]

N[μ,1000]

zero=LogIntegral[Root[{LogIntegral[#1]&,1.4513692348833810502839684858920274104826063443488632576253`30.102999566398122}]]

Simplify[zero]

FullSimplify[zero]

N[zero]

N[zero,20]

BBP-Type

Search

FindRoot[LogIntegral[x],{x,1.4},WorkingPrecision->1000]

μ=Root[{LogIntegral[#1]&,1.4513692348833810502839684858920274104826063443488632576253`30.102999566398122}];

<<Utilities`Simplify`

Monitor[Do[If[(b=FindBasis[μ,-6^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-6^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[μ,-5^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-5^k,n,p}];Print["DONE"]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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