A105169 - OEIS

A105169

Decimal expansion of ultraradical of Pi: real x such that x^5 + x = Pi.

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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 on the unsolvability of the general quintic. The constant given here results from numerical evaluation of the irreducible quintic equation x^5 + x = Pi.

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]

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.

Index entries for transcendental numbers

FORMULA

The decimal expansion of Pi is given in A000796.

EXAMPLE

1.1479653855...

MATHEMATICA

RealDigits[x/.FindRoot[x^5+x==Pi, {x, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jun 05 2016 *)