A318732 - OEIS

%I #26 Jan 13 2024 16:28:40

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

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

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

%N Decimal expansion of the nontrivial real solution to x^6 - x^3 + x^2 + 2*x - 1 = 0.

%C The first part of Ramanujan's question 699 in the Journal of the Indian Mathematical Society (VII, 199) asked "Show that the roots of the equations x^6 - x^3 + x^2 + 2*x - 1 = 0, ... can be expressed in terms of radicals."

%C The polynomial includes a trivial factor, i.e., x^6 - x^3 + x^2 + 2*x - 1 = (x + 1) * (x^5 - x^4 + x^3 - 2*x^2 + 3*x - 1).

%D V. M. Galkin, O. R. Kozyrev, On an algebraic problem of Ramanujan, pp. 89-94 in Number Theoretic And Algebraic Methods In Computer Science - Proceedings Of The International Conference, Moscow 1993, Ed. Horst G. Zimmer, World Scientific, 31 Aug 1995

%H B. C. Berndt, Y. S. Choi, S. Y. Kang, <a href="https://faculty.math.illinois.edu/~berndt/jims.ps">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q699, JIMS VII).

%F Equals 2^(1/4) / G(79), where G(n) is Ramanujan's class invariant G(n) = 2^(-1/4) * q(n)^(-1/24) * Product_{k>=0} (1 + q(n)^(2*k + 1)), with q(n) = exp(-Pi * sqrt(n)).

%e 0.441804263270765321567119439396889005149374940909247541777660...

%t RealDigits[Root[x^6-x^3+x^2+2x-1,2],10,120][[1]] (* _Harvey P. Dale_, Jan 13 2024 *)

%o (PARI) p(x)=x^5-x^4+x^3-2*x^2+3*x-1;solve(x=0.3,0.5,p(x))

%o (PARI) q(x)=exp(-Pi*sqrt(x)); G(n)=2^(-1/4)*q(n)^(-1/24)*prodinf(k=0,(1+q(n)^(2*k+1))); 2^(1/4)/G(79)

%Y Cf. A318733.

%K nonn,cons

%O 0,1

%A _Hugo Pfoertner_, Sep 02 2018