A294644 - OEIS

%I #35 Oct 27 2023 10:30:39

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

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

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

%N Decimal expansion of the real positive solution to x^3 = x + 3.

%C From _Wolfdieter Lang_, Sep 13 2022: (Start)

%C The other two solutions are (w1*(81/2 + (3/2)*sqrt(717))^(1/3) + w2*(81/2 - (3/2)*sqrt(717))^(1/3))/3 = -0.835849940... + 1.04686931...*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.

%C With hyperbolic functions these solutions are -(1/3)*sqrt(3)*(cosh((1/3)*arccosh((9/2)*sqrt(3))) - sqrt(3)*sinh((1/3)*arccosh((9/2)*sqrt(3)))*i), and its complex conjugate.(End)

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

%F Equals (3 + (3 + (3 + ...)^(1/3))^(1/3))^(1/3).

%F Equals (3/2 + sqrt(239/108))^(1/3) + (3/2 - sqrt(239/108))^(1/3).

%F From _Wolfdieter Lang_, Sep 13 2022: (Start)

%F Equals (1/6)*(324 + 12*sqrt(717))^(1/3) + 2/(324 + 12*sqrt(717))^(1/3).

%F Equals ((81/2 + (3/2)*sqrt(717))^(1/3) + (81/2 - (3/2)*sqrt(717))^(1/3))/3.

%F Equals (2/3)*sqrt(3)*cosh((1/3)*arccosh((9/2)*sqrt(3))). (End)

%e 1.6716998816571609697481497812195572287...

%t RealDigits[x /. FindRoot[x^3 - x - 3, {x, 1}, WorkingPrecision -> 120], 10, 100][[1]] (* _Amiram Eldar_, Jun 01 2021 *)

%o (PARI) solve(n=0,2,n^3-n-3)

%o (PARI) polrootsreal(x^3-x-3)[1] \\ _Charles R Greathouse IV_, Nov 05 2017

%o (MATLAB) solve('x^3=x+3');ans(1),eval(ans),  %  _Michal Paulovic_, Mar 08 2021

%Y Cf. A337569.

%K nonn,cons

%O 1,2

%A _Michal Paulovic_, Nov 05 2017