A105172 Ultraradical of phi: decimal expansion of the real x such that x^5 + x = phi.

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

Offset

0,1

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 = phi.

References

Birkhoff, G. and Mac Lane, S. "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]

S. R. Finch, "The Golden Mean." Section 1.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 5-12, 2003.

Links

Table of n, a(n) for n=0..99.

Eric Weisstein's World of Mathematics, Ultraradical.

Eric Weisstein's World of Mathematics, Quintic Equation.

Eric Weisstein's World of Mathematics, Golden Ratio.

Formula

The decimal expansion of phi, the golden ratio, is given in A001622.

Example

0.928380799258974

Crossrefs

Cf. A001622.

Sequence in context: A299957 A252001 A098784 * A231533 A011453 A125580

Adjacent sequences: A105169 A105170 A105171 * A105173 A105174 A105175

Keyword

cons,nonn

Author

Jonathan Vos Post, Apr 11 2005

Status

approved