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Decimal expansion of the positive real root of x^4 + x - 1.
2

%I #25 Oct 15 2024 04:57:54

%S 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,

%T 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,

%U 2,9,0,6,4,7,5,1

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

%C The other real (negative) root is -A060007.

%C 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.

%F 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))).

%e r = 0.724491959000515611588372282187036565786494481350011017270...

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

%o (PARI) solve(x=0, 1, x^4 + x - 1) \\ _Michel Marcus_, Aug 28 2022

%Y Cf. A060007, A376658.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Aug 27 2022