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%I #34 Sep 16 2018 16:39:12
%S 5,7,6,4,7,1,4,2,9,6,1,9,5,5,0,6,1,0,4,8,6,3,5,4,4,0,0,1,7,7,5,7,8,5,
%T 1,7,4,7,7,3,4,2,1,8,2,1,6,1,4,7,9,0,4,9,5,3,1,2,0,0,5,8,8,4,2,6,1,1,
%U 8,7,9,3,3,9,2,6,3
%N Decimal expansion of the nontrivial real solution to x^6 + x^5 - x^3 - x^2 - x + 1 = 0.
%C The second 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^5 - x^3 - x^2 - x + 1 = 0 can be expressed in terms of radicals."
%C The polynomial includes a trivial factor, i.e., x^6 + x^5 - x^3 - x^2 - x + 1 = (x - 1) * (x^5 + 2*x^4 + 2*x^3 + x^2 - 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 Expressed in radicals, the number is
%F (1/20)*4^(4/5)*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(1/5) - (329*sqrt(5)/sqrt(235 + 94*sqrt(5)) - 57*sqrt(5) + 9*sqrt(235 + 94*sqrt(5)) - 89)*4^(3/5)/(20*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(3/5)) - (47*sqrt(5)/sqrt(235 + 94*sqrt(5)) + 23*sqrt(5) - 3*sqrt(235 + 94*sqrt(5)) - 3)* 4^(2/5)/(20*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(2/5)) + (-1 + 2*sqrt(5))*4^(1/5)/(5*((215*sqrt(5)*sqrt(235 + 94*sqrt(5)) - 10575 - 5405*sqrt(5) + 597*sqrt(235 + 94*sqrt(5)))/sqrt(235 + 94*sqrt(5)))^(1/5)) - 2/5. - _Robert Israel_, Sep 04 2018
%F Equals 2^(1/4) / G(47), 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)). - _Hugo Pfoertner_, Sep 15 2018
%e 0.5764714296195506104863544001775785174773421821614790...
%o (PARI) p(x)=x^5+2*x^4+2*x^3+x^2-1; solve(x=0.3,0.7,p(x))
%Y Cf. A318732.
%K nonn,cons
%O 0,1
%A _Hugo Pfoertner_, Sep 02 2018
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