OFFSET
0,1
COMMENTS
The other roots are w1*(4*(9 + sqrt(85)))^(1/3) + ((4*(9 - sqrt(85)))^(1/3)))/6 = -0.2682825823... + 0.741120749...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (-sinh((1/3)*arcsinh(9/2)) + sqrt(3)*cosh((1/3)*arcsinh(9/2))*i)/3, and its complex conjugate.
FORMULA
r = ((4*(9 + sqrt(85)))^(1/3) - 4*(4*(9 + sqrt(85)))^(-1/3))/6.
r = ((4*(9 + sqrt(85)))^(1/3) + w1*((4*(9 - sqrt(85)))^(1/3)))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (2/3)*sinh((1/3)*arcsinh(9/2)).
EXAMPLE
0.5365651646722229187574245122387738338212422637521880663142371514206...
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