A318732 Decimal expansion of the nontrivial real solution to x^6 - x^3 + x^2 + 2*x - 1 = 0.

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

Offset

0,1

Comments

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."

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).

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

Table of n, a(n) for n=0..86.

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).

Formula

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)).

Example

0.441804263270765321567119439396889005149374940909247541777660...

Mathematica

RealDigits[Root[x^6-x^3+x^2+2x-1, 2], 10, 120][[1]] (* Harvey P. Dale, Jan 13 2024 *)

Prog

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

Crossrefs

Cf. A318733.

Sequence in context: A128213 A171716 A211788 * A016706 A358204 A138679

Adjacent sequences: A318729 A318730 A318731 * A318733 A318734 A318735

Keyword

nonn,cons

Author

Hugo Pfoertner, Sep 02 2018

Status

approved