A273066 Decimal expansion of the real root of x^3 - 2x + 2, negated.

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

Offset: 1

Author

Rick L. Shepherd, May 14 2016

Keywords

• nonn
• cons

Comments

The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. The only other real, cubic, monic polynomial with nonzero constant term and equal coefficients and roots when ignoring the leading coefficient is x^3 + x^2 - x - 1 (per the Math Overflow link).

The other two roots of x^3 - 2*x - 2 are w1*(1 + (1/9)*sqrt(57))^(1/3) + w2*(1 - (1/9)*sqrt(57))^(1/3) = -0.88464617... + 0.58974280...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1. Using hyperbolic functions this is -(1/3)*sqrt(6)*(cosh((1/3)*arccosh((3/4)*sqrt(6))) - sqrt(3)*sinh((1/3)*arccosh((3/4)*sqrt(6)))*i) and its complex conjugate. - Wolfdieter Lang, Sep 13 2022

Examples

			1.7692923542386314152404094643350334926705530458988570042331061304026738...
		

Links

• A. J. Di Scala and O. Macia, Finiteness of Ulam Polynomials, arXiv:0904.0133 [math.AG], 2009.
• R. Stanley, R. Israel et al., Math Overflow: Which polynomial's roots are its coefficients?, Sep 3 2015.
• P. R. Stein, On Polynomial Equations with Coefficients Equal to Their Roots, The American Mathematical Monthly, Vol. 73, No. 3 (Mar., 1966), pp. 272-274.
• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A273065, A273067, A357102.

Programs

Mathematica

RealDigits[Root[x^3-2x+2,1],10,120][[1]] (* Harvey P. Dale, May 25 2022 *)

PARI

default(realprecision, 200);
-solve(x = -1.8, -1.7, x^3 - 2*x + 2)

Formula

Equals ((9-sqrt(57))^(1/3))/(3^(2/3)) + 2/((3(9-sqrt(57)))^(1/3)) (from Wolfram Alpha).

From Wolfdieter Lang, Sep 13 2022: (Start)

Equals (1 + (1/9)*sqrt(57))^(1/3) + (2/3)*(1 + (1/9)*sqrt(57))^(-1/3) [compare with the above formula which uses the negative sqrt(57)].

Equals (1 + (1/9)*sqrt(57))^(1/3) + (1 - (1/9)*sqrt(57))^(1/3).

Equals (2/3)*sqrt(6)*cosh((1/3)*arccosh((3/4)*sqrt(6))). (End)