Baxter's Four-Coloring Constant

Author

Eric W. Weisstein
July 11, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/BaxtersFour-ColoringConstant.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

General

c=FullSimplify[-(1/x)Product[(((1-(x^(6j-3)))(((1-(x^(6j-2))))^2)(1-(x^(6j-1))))/((1-(x^(6j-5)))(1-(x^(6j-4)))(1-(x^(6j)))(1-(x^(6j+1))))),{j,∞}]]

TraditionalForm[c]

Solve[q==2-x-(x^(-1)),x]//FullSimplify

expr=c/.x->(1/2) (2+(Sqrt[-4+q]) (Sqrt[q])-q);

Plot[expr,{q,4,5}]

q==4

expr/.q->4.0001

Solve[4==2-x-(x^(-1)),x]//FullSimplify

N[3/(4 Pi^2) Gamma[1/3]^3,20]

Limit[expr,q->4]//Timing

Limit[expr,q->4,Direction->-1]//Timing

Series[expr,{q,4,0}]

FullSimplify[%]

Value

C2=Product[((((3j-1))^2)/((3j-2)(3j))),{j,∞}]

FullSimplify[C2-(3/(4(π^2)))((Gamma[(1/3)])^3)]

N[C2-(3/(4(π^2)))((Gamma[(1/3)])^3),10^2]

PossibleZeroQ[%]

<<Utilities`Simplify`

GammaSimplify[C2]//FullSimplify

N[%]

N[(3/(4(π^2)))((Gamma[(1/3)])^3),10]