Mills' Constant

Author

Eric W. Weisstein
January 24, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Sequences

Caldwell and Cheng (2005)

a[n_]:={3,30,6,80,12,450,894,3636,70756,97220,66768}[[n]]
b[1]:=2
b[n_]:=b[n]=b[n-1]^3+a[n-1]

Search

Using NextPrime

<<NumberTheory`NumberTheoryFunctions`

b=2^3;
Do[
Print[{n,Timing[(p=(np=NextPrime[b])-b);b=np^3][[1]],p}],
{n,10}
]

By hand (should be the same); gives access to p for partial runs

b=2^3;
Do[
p=1;
Print[{n,Timing[While[!PrimeQ[b+p],p++];b=(b+p)^3][[1]],p}],
{n,10}
]

Value

Table[Block[{$MaxExtraPrecision=Infinity},
N[Subtract@@{((b[n])^(3^(-n))),(((b[n]+1))^(3^(-n)))},10^5]
]//N,{n,12}]

N[θ=With[{n=12},((b[n])^(3^(-n)))],1000]

Block[{$MaxExtraPrecision=Infinity},Table[Floor[(θ^(3^n))],{n,5}]]//Timing

Block[{$MaxExtraPrecision=Infinity},Table[Floor[(θ^(3^n))]==b[n],{n,12}]]//Timing