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A257240
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Decimal expansion of the real root of x^3 - 3*x - 10.
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0
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2, 6, 1, 2, 8, 8, 7, 8, 6, 4, 7, 1, 7, 5, 4, 4, 7, 5, 4, 4, 0, 7, 2, 4, 9, 9, 3, 8, 6, 2, 9, 7, 6, 2, 9, 1, 2, 8, 7, 5, 7, 7, 1, 2, 8, 4, 8, 0, 6, 3, 2, 8, 1, 7, 2, 3, 0, 2, 7, 0, 0, 5, 1, 8, 2, 1, 0, 1, 8, 3, 5, 8, 4, 9, 1, 1, 2, 5, 7, 3, 6, 3, 4, 4, 2, 2, 7, 1, 1, 3, 9, 6, 0, 1, 9, 8, 4, 8, 5, 6, 8, 6, 7, 6, 0, 3, 6, 8, 1, 9, 0, 6, 1, 3, 2, 0, 6, 7, 5, 6, 3, 7, 2, 8, 3, 9, 8, 7, 4
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OFFSET
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1,1
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COMMENTS
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This is related to the fourth of thirty problems posed by Niccolò Tartaglia to Antonio Maria Fiore in the year 1535 (in Venice it was still 1534). See the Katscher reference [in German] pp. 14, 15.
The problem is: find me a number which when 3 of its cubic roots are subtracted leaves 10. That is z - 3*z^(1/3) = 10, or, with z = x^3, x^3 - 3*x = 10, with real solution x1. The solution to the problem is then z1 = x1^3 = 13 - 4*x1 (see the example section).
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REFERENCES
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Friedrich Katscher, Die Kubischen Gleichungen bei Nicolo Tartaglia, Verlag der Österreichischen Akademie der Wissenschaften, 2001, Wien, Aufgabe XXV, pp. 13-16.
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LINKS
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FORMULA
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The real solution x1 to x^3 - 3*x - 10 = 0 is
x1 = (5 + 2*sqrt(6))^(1/3) + (5 - 2*sqrt(6))^(1/3).
The two complex solutions are a + b*i and a - b*i, with a = -x1/2 and b = sqrt(3)*y1/2 where y1 = (5 + 2*sqrt(6))^(1/3) - (5 - 2*sqrt(6))^(1/3).
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EXAMPLE
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x1 = 2.6128878647175447544072499386297629...
y1 = 1.6814229074174677895820170587695490...
z1 = 17.8386635941526342632217498158892887...
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MATHEMATICA
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RealDigits[Root[x^3-3x-10, 1], 10, 130][[1]] (* Harvey P. Dale, Dec 13 2021 *)
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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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