OFFSET
0,1
COMMENTS
The other (complex) roots are (w1*(4*(9 + sqrt(77)))^(1/3) + w2*(4*(9 - sqrt(77)))^(1/3))/6 = -0.4256915364... + 0.458591887...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3))/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (-cosh((1/3)*arccosh(9/2)) + sqrt(3)*sinh((1/3)*arccosh(9/2))*i)/3, and its complex conjugate.
FORMULA
r = ((4*(9 + sqrt(77)))^(1/3) + 4*(4*(9 + sqrt(77)))^(-1/3))/6.
r = ((4*(9 + sqrt(77)))^(1/3) + (4*(9 - sqrt(77)))^(1/3))/6.
r = (2/3)*cosh((1/3)*arccosh(9/2)).
EXAMPLE
0.851383072866924393493940112187859385096149923938041965059002396279722...
MATHEMATICA
RealDigits[x /. FindRoot[3*x^3 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 17 2022
STATUS
approved