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Decimal expansion of the real root of x^3 - 2*x^2 - 2.
1

%I #20 Oct 27 2023 10:38:10

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

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

%U 7,5,5,8,8,9,4,4,6,5

%N Decimal expansion of the real root of x^3 - 2*x^2 - 2.

%C This equals r0 + 2/3 where r0 is the real root of y^3 - (4/3)*y - 70/27.

%C The other two roots are (w1*(35 + 3*sqrt(129))^(1/3) + w2*(35 - 3*sqrt(129))^(1/3 + 2)/3 = -0.1796520429... + 0.9030131458...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

%C Using hyperbolic functions these roots are (2/3)*(1 - cosh((1/3)*arccosh(35/8)) + sqrt(3)*sinh((1/3)*arccosh(35/8))*i), and its complex conjugate.

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

%F r = ((35 + 3*sqrt(129))^(1/3) + 4*(35 + 3*sqrt(129))^(-1/3) + 2)/3.

%F r = ((35 + 3*sqrt(129))^(1/3) + (35 - 3*sqrt(129))^(1/3) + 2)/3.

%F r = (2/3)*(2*cosh((1/3)*arccosh(35/8)) + 1).

%e 2.359304085971776420730660392800532555346481278067672283797141251583875588...

%p h := ((35 + 3*sqrt(129))/8)^(1/3): evalf((1 + h + 1/h)*2/3, 82); # _Peter Luschny_, Sep 25 2022

%t RealDigits[x /. FindRoot[x^3 - 2*x^2 - 2, {x, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Sep 21 2022 *)

%o (PARI) polrootsreal(x^3 - 2*x^2 - 2)[1] \\ _Michel Marcus_, Sep 23 2022

%Y Cf. A058265 - 1 (x^3 + 2*x^2 - 2).

%K nonn,cons,easy

%O 1,1

%A _Wolfdieter Lang_, Sep 20 2022