Plastic Constant

Author

Eric W. Weisstein
March 20, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Value

p=ExactConstant["plastic constant"]

Simplify[ToRadicals[p]]

N[p,20]

p2=With[{r=-(1/5)(((-1728KleinInvariantJ[(1/2)(1+I Sqrt[23])]))^(1/3))},((11r+52)/(5r-61))]

FullSimplify[p2]

N[{p,p2},20]

Limit

ratio=(((Root[-1-#1+((#1)^3)&,1])^n) Root[-1+6 #1-23 ((#1)^2)+23 ((#1)^3)&,1]+((Root[-1-#1+((#1)^3)&,3])^n) Root[-1+6 #1-23 ((#1)^2)+23 ((#1)^3)&,2]+((Root[-1-#1+((#1)^3)&,2])^n) Root[-1+6 #1-23 ((#1)^2)+23 ((#1)^3)&,3])/(((Root[-1-#1+((#1)^3)&,2])^n) Root[-1-5 #1+23 ((#1)^3)&,2]+((Root[-1-#1+((#1)^3)&,1])^(-1+n)) Root[-1+6 #1-23 ((#1)^2)+23 ((#1)^3)&,1]+((Root[-1-#1+((#1)^3)&,3])^(-1+n)) Root[-1+6 #1-23 ((#1)^2)+23 ((#1)^3)&,2]);

p=Limit[ratio,n->∞]

Nested Radical

FixedPoint[(((1+#))^(1/3))&,1.]

FixedPoint[(((1+#))^(1/3))&,1`10]

FixedPoint[(((1+#))^(1/3))&,1`100,SameTest->(Abs[#1-#2]<10^-90&)]

FixedPoint[N[(((1+#))^(1/3)),20]&,1]

FixedPoint[N[(((1+#))^(1/3)),50]&,1]

Precision[(((1+1`20))^(1/3))]

(P^24)

q=-With[{τ=(1/2)(1+I Sqrt[23])},(((((DedekindEta[τ])/((Sqrt[2])DedekindEta[2τ]))))^24)]

...wrong...

N[{(p^24),q},20]//Chop

Approximations

N[Subtract@@{Exp[π Sqrt[23]],(2^12)(p^24)-24},20]

Algebraic Identities

Reduce[P-1==(P^(-4)),P,Reals]

Reduce[P+1==(P^3),P,Reals]

Only the golden ratio and P satisfy the following identities

Reduce[x+1==(x^k)&&x-1==(x^(-l))&&(k|l)∈Integers&&k>0&&l>0,{x,k,l},Reals]