Goldbach Conjecture

Author

Eric W. Weisstein
January 14, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/GoldbachConjecture.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Extended Goldbach Conjecture

p(n)=(∏_({{k=2}, {(Subscript[p, k])|n}}))(((Subscript[p, k])-1)/((Subscript[p, k])-2))

<<NumberTheory`NumberTheoryFunctions`

prod[n_] := Module[{v =Rest[PrimeFactorList[2 n]]},
Times @@ (v - 2)/Times @@ (v - 1)]

f[n_] := Length[Select[2n - Prime[Range[PrimePi[n]]], PrimeQ]]

ListPlot[Table[{2 n, Log[2 n, f[n] prod[n]]},{n,4,10000}],
     PlotRange->{0,0.6},PlotStyle-> {Red,AbsolutePointSize[1]},
AxesLabel->]

Numerical Values of Limits

N[3^3^15]

N[3^3^15,20]

N[E^E^11.503]

N[E^E^11.503,20]