A105172 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

References

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

Steven 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.

Carl Runge, Über die auflösbaren Gleichungen von der Form x^5 + ux + v = 0, Acta Math. 7 (1885), 173-186. [in German]

Eric Weisstein's World of Mathematics, Golden Ratio.

Eric Weisstein's World of Mathematics, Quintic Equation.

Eric Weisstein's World of Mathematics, Ultraradical.

Formula

The minimal polynomial is x^10 + 2*x^6 - x^5 + x^2 - x - 1. - Joerg Arndt, Apr 12 2026

Example

0.92838079925897402951465604466120701517783700628447...

Mathematica

RealDigits[x /. FindRoot[x^5 + x - GoldenRatio, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Apr 12 2026 *)

Crossrefs

Cf. A001622 (phi).

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