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 A257240 Decimal expansion of the real root of x^3 - 3*x - 10. 0
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This is related to the fourth of thirty problems posed by NiccolĂ˛ 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). REFERENCES Friedrich Katscher, Die Kubischen Gleichungen bei Nicolo Tartaglia, Verlag der Ă–sterreichischen Akademie der Wissenschaften, 2001, Wien, Aufgabe XXV, pp. 13-16. LINKS MacTutor History of Mathematics, Nicolo Tartaglia. FORMULA 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). EXAMPLE x1 = 2.6128878647175447544072499386297629... y1 = 1.6814229074174677895820170587695490... z1 = 17.8386635941526342632217498158892887... PROG (PARI) polrootsreal(x^3-3*x-10)[1] \\ Charles R Greathouse IV, May 21 2015 CROSSREFS Cf. A257235, A257236, A257237, A257239. Sequence in context: A078434 A021892 A269224 * A121601 A122761 A100469 Adjacent sequences:  A257237 A257238 A257239 * A257241 A257242 A257243 KEYWORD nonn,easy,cons AUTHOR Wolfdieter Lang, May 21 2015 STATUS approved

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Last modified July 23 00:43 EDT 2019. Contains 325228 sequences. (Running on oeis4.)