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A318732
Decimal expansion of the nontrivial real solution to x^6 - x^3 + x^2 + 2*x - 1 = 0.
2
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
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
KEYWORD
nonn,cons
AUTHOR
Hugo Pfoertner, Sep 02 2018
STATUS
approved