A356032 Decimal expansion of the positive real root of x^4 + x - 1.

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

Offset: 0

Author

Wolfdieter Lang, Aug 27 2022

Keywords

• nonn
• cons
• easy

Comments

The other real (negative) root is -A060007.

One of the pair of complex conjugate roots is obtained by negating sqrt(2*u) and sqrt(u) in the formula for r below, giving 0.248126062... - 1.033982060...*i.

Also, the absolute value of the negative real root of x^4 - x - 1, cf. A060007. - M. F. Hasler, Jul 12 2025

Examples

			r = 0.724491959000515611588372282187036565786494481350011017270...
		

Crossrefs

Cf. A060007 (positive root of x^4 - x - 1), A072223, A086106, A202538, A376658.

Programs

Mathematica

First[RealDigits[x/.N[{x->Root[-1+#1+#1^4 &,2,0]},75]]] (* Stefano Spezia, Aug 27 2022 *)

PARI

solve(x=0, 1, x^4 + x - 1) \\ Michel Marcus, Aug 28 2022

PARI

polrootsreal(x^4 + x - 1)[2] \\ M. F. Hasler, Jul 12 2025

Formula

r = (-sqrt(2)*u + sqrt(sqrt(2*u) - 2*u^2))/(2*sqrt(u)), with u = (Ap^(1/3) + ep*Am^(1/3))/3, where Ap = (3/16)*(9 + sqrt(3*283)), Am = (3/16)*(9 - sqrt(3*283)) and ep = (-1 + sqrt(3)*i)/2, with i = sqrt(-1). For the trigonometric version set u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/16)* sqrt(3))).

Equals sqrt(A072223) = 1/A086106 = 1/exp(A202538). - Hugo Pfoertner, Jul 13 2025