OFFSET
1,2
COMMENTS
x = (3 - (3 - (3 - ...)^(1/3))^(1/3))^(1/3).
The other two solutions are (w1*(81/2 + (3/2)*sqrt(741))^(1/3) + (81/2 - (3/2)*sqrt(741))^(1/3))/3 = -0.60670583... + 1.45061225...*i, where w1 = (-1 + sqrt(3)*i)/2, and its complex conjugate. With hyperbolic functions these solutions are -(1/3)*sqrt(3)*(sinh((1/3)*arcsinh((9/2)*sqrt(3))) - sqrt(3)*cosh((1/3)*arcsinh((9/2)*sqrt(3)))*i), and its complex conjugate. - Wolfdieter Lang, Sep 13 2022
FORMULA
Equals (3/2 + sqrt(741/324))^(1/3) - (-3/2 + sqrt(741/324))^(1/3).
From Wolfdieter Lang, Sep 13 2022: (Start)
Equals (1/6)*(324 + 12*sqrt(741))^(1/3) - 2/(324 + 12*sqrt(741))^(1/3).
Equals ((81/2 + (3/2)*sqrt(741))^(1/3) + w1*(81/2 - (3/2)*sqrt(741))^(1/3))/3, with w1 = (-1 + sqrt(3)*i)/2, one of the complex roots of x^3 - 1.
Equals (2/3)*sqrt(3)*sinh((1/3)*arcsinh((9/2)*sqrt(3))). (End)
EXAMPLE
1.2134116627622296...
MAPLE
Digits:=100; solve(x^3+x-3=0); evalf(%)[1];
MATHEMATICA
RealDigits[x /. FindRoot[x^3 + x - 3, {x, 1}, WorkingPrecision -> 100], 10, 90][[1]] (* Amiram Eldar, Sep 03 2020 *)
PROG
(PARI) solve(n=0, 2, n^3+n-3]
(PARI) polroots(n^3+n-3)[1]
(PARI) polrootsreal(x^3+x-3)[1] \\ Charles R Greathouse IV, Oct 27 2023
(MATLAB) format long; solve('x^3+x-3=0'); ans(1), (eval(ans))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Michal Paulovic, Sep 01 2020
STATUS
approved