A337569 Decimal expansion of the real solution to x^3 = 3 - x.

1, 2, 1, 3, 4, 1, 1, 6, 6, 2, 7, 6, 2, 2, 2, 9, 6, 3, 4, 1, 3, 2, 1, 3, 1, 3, 7, 7, 3, 8, 1, 4, 8, 9, 5, 2, 6, 6, 2, 2, 7, 0, 6, 5, 7, 3, 9, 6, 9, 8, 9, 3, 4, 9, 5, 5, 2, 7, 5, 6, 8, 3, 6, 2, 4, 2, 5, 6, 3, 2, 6, 9, 5, 2, 7, 7, 3, 8, 6, 9, 1, 7, 4, 0, 3, 5, 9, 2, 1, 3, 9, 1, 8, 4, 4, 4, 1

Offset: 1

Author

Michal Paulovic, Sep 01 2020

Keywords

• nonn
• cons

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

Examples

			1.2134116627622296...
		

Links

• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A294644, A357102.

Programs

MATLAB

format long; solve('x^3+x-3=0'); ans(1), (eval(ans))

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 *)

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

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)