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A105172
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Ultraradical of phi: decimal expansion of the real x such that x^5 + x = phi.
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0
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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.
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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, "Ueber 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.
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LINKS
| Eric Weisstein's World of Mathematics, Ultraradical.
Eric Weisstein's World of Mathematics, Quintic Equation.
Eric Weisstein's World of Mathematics, Golden Ratio.
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FORMULA
| The decimal expansion of phi, the golden ratio, is given in A001622.
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EXAMPLE
| 0.928380799258974
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CROSSREFS
| Cf. A001622.
Sequence in context: A111722 A137301 A098784 * A011453 A125580 A086238
Adjacent sequences: A105169 A105170 A105171 * A105173 A105174 A105175
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KEYWORD
| cons,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 11 2005
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