A318733 Decimal expansion of the nontrivial real solution to x^6 + x^5 - x^3 - x^2 - x + 1 = 0.

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, 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, 8, 7, 9, 3, 3, 9, 2, 6, 3

Offset: 0

Author

Hugo Pfoertner, Sep 02 2018

Keywords

• nonn
• cons

Comments

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."

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).

Examples

			0.5764714296195506104863544001775785174773421821614790...
		

References

• 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

Links

• B. C. Berndt, Y. S. Choi, S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q699, JIMS VII).

Crossrefs

Cf. A318732.

Programs

PARI

p(x)=x^5+2*x^4+2*x^3+x^2-1; solve(x=0.3,0.7,p(x))

Formula

Expressed in radicals, the number is

(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

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