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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 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 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 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 + 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][[1]] (* Robert G. Wilson v, May 22 2015 *) PROG (PARI) polrootsreal(x^3+4*x-13)[1] \\ 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.)