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
Eric Weisstein's World of Mathematics, Ultraradical.
Eric Weisstein's World of Mathematics, Quintic Equation.
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 *)
PROG
(PARI) solve(x=1, 2, x^5+x-Pi) \\ Charles R Greathouse IV, Oct 01 2022
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Jonathan Vos Post, Apr 11 2005
STATUS
approved