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A105171
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Ultraradical of e. Decimal expansion of the real x such that x^5 + x = e.
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0
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1, 1, 0, 0, 9, 3, 2, 6, 6, 5, 1, 9, 3, 6, 2, 6, 6, 0, 7, 6, 5, 4, 9, 7, 5, 8, 8, 0, 2, 6, 1, 4, 0, 8, 3, 6, 0, 8, 4, 1, 1, 8, 4, 8, 2, 8, 0, 1, 9, 4, 6, 5, 1, 6, 1, 8, 1, 3, 6, 2, 0, 7, 4, 5, 6, 8, 5, 9, 9, 6, 8, 1, 4, 5, 2, 0, 6, 2, 4, 9, 7, 6, 1, 7, 1, 2, 5, 2, 1, 4, 4, 8, 5, 7, 0, 5, 0, 1, 6, 1
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OFFSET
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1,5
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COMMENTS
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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 = e.
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REFERENCES
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G. Birkhoff and S. Mac Lane, "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]
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LINKS
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Eric Weisstein's World of Mathematics, e
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FORMULA
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The decimal expansion of e is given in A001113.
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EXAMPLE
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1.10093266519...
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MATHEMATICA
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RealDigits[x/.FindRoot[x^5+x==E, {x, 1.1}, WorkingPrecision->150]][[1]] (* Harvey P. Dale, Mar 09 2011 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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