|
%I #7 Jun 23 2014 16:31:12
%S 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,
%T 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,
%U 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
%N Ultraradical of phi: decimal expansion of the real x such that x^5 + x = phi.
%C 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.
%D 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.
%D C. Runge, "Über die aufloesbaren Gleichungen von der Form x^5 + ux + v = 0", Acta Math. 7, 173-186, 1885. [German]
%D S. R. Finch, "The Golden Mean." Section 1.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 5-12, 2003.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Ultraradical.html">Ultraradical</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuinticEquation.html">Quintic Equation.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio.</a>
%F The decimal expansion of phi, the golden ratio, is given in A001622.
%e 0.928380799258974
%Y Cf. A001622.
%K cons,nonn
%O 0,1
%A _Jonathan Vos Post_, Apr 11 2005
|
