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 A257239 Decimal expansion of the real root of x^3 + 4*x - 13. 1
 1, 7, 9, 7, 6, 6, 5, 4, 9, 4, 4, 0, 0, 4, 6, 1, 4, 6, 0, 9, 8, 9, 1, 6, 1, 9, 4, 3, 0, 6, 0, 2, 3, 6, 4, 6, 1, 3, 4, 0, 4, 3, 3, 6, 9, 3, 3, 5, 1, 8, 4, 3, 4, 3, 1, 7, 5, 7, 8, 9, 9, 5, 1, 2, 3, 9, 2, 2, 5, 2, 4, 8, 0, 8, 4, 9, 4, 0, 0, 0, 9, 9, 9, 3, 7, 8, 6, 1, 7, 3, 6, 5, 0, 2, 9, 2, 2, 8, 1, 2, 3, 7, 5, 2, 2 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This is related to the third 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 added to 4 times its cube root gives 13. That is z + z^(1/3) = 13, or, with z = x^3, x^3 + 4*x = 13, with real solution x1. The solution to the problem is then z1 = x1^3 = 13 - 4*x1 (see the formula and example section). REFERENCES Friedrich Katscher, Die Kubischen Gleichungen bei Nicolo Tartaglia, Verlag der Österreichischen 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 + 4*x - 13 = 0 is x1 = (1/6)*((1404 + 12*sqrt(14457))^(1/3) - (-1404 + 12*sqrt(14457))^(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 = (1/6)*((1404+12*sqrt(14457))^(1/3) + (-1404 + 12*sqrt(14457))^(1/3)) with y1 = 2.926590945638182088730632869966915335446... and z1 = 5.809338022398154156043352227759054154638... EXAMPLE x1 = 1.797665494400461460989161943060236461340... MATHEMATICA RealDigits[ Solve[x^3 + 4*x - 13 == 0, x][[1, 1, 2]], 10, 111][] (* Robert G. Wilson v, May 22 2015 *) PROG (PARI) polrootsreal(x^3+4*x-13) \\ Charles R Greathouse IV, May 21 2015 CROSSREFS Cf. A257235, A257236, A257237. Sequence in context: A177271 A188157 A135000 * A199742 A258112 A076668 Adjacent sequences:  A257236 A257237 A257238 * A257240 A257241 A257242 KEYWORD nonn,easy,cons AUTHOR Wolfdieter Lang, May 21 2015 STATUS approved

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Last modified March 21 21:34 EDT 2019. Contains 321382 sequences. (Running on oeis4.)