A105171 Ultraradical of e. Decimal expansion of the real x such that x^5 + x = e.

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

Offset

1,5

Comments

Weisstein explains a term apparently coined by Ian Stewart: "Ultraradical: A symbol which can be used to express solutions not obtainable by finite root extraction. The solution to the irreducible quintic equation x^5 + x = a" can be written Ultraradical(a). We know from the classic papers by Abel and Galois of the unsolvability of the general quintic. The constant given here results from numerical evaluation of the irreducible quintic equation x^5 + x = e.

References

G. Birkhoff and S. Mac Lane, "Insolvability of Quintic Equations." Section 15.8 in A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 418-421, 1996.

C. Runge, "Über die aufloesbaren Gleichungen von der Form x^5 + ux + v = 0", Acta Math. 7, 173-186, 1885. [German]

Links

Table of n, a(n) for n=1..100.

Eric Weisstein's World of Mathematics, Ultraradical

Eric Weisstein's World of Mathematics, Quintic Equation

Eric Weisstein's World of Mathematics, e

Wikipedia, Bring radical

Formula

The decimal expansion of e is given in A001113.

Example

1.10093266519...

Mathematica

RealDigits[x/.FindRoot[x^5+x==E, {x, 1.1}, WorkingPrecision->150]][[1]]  (* Harvey P. Dale, Mar 09 2011 *)
RealDigits[ Root[ #^5 + # - E&, 1], 10, 100] // First (* Jean-François Alcover, Feb 27 2013 *)

Prog

(PARI) solve(x=1, 2, x^5+x-exp(1)) \\ Charles R Greathouse IV, May 28 2014

Crossrefs

Cf. A001113.

Sequence in context: A072559 A019941 A200624 * A369880 A010538 A216102

Adjacent sequences: A105168 A105169 A105170 * A105172 A105173 A105174

Keyword

cons,nonn

Author

Jonathan Vos Post, Apr 11 2005

Status

approved