A257239 Decimal expansion of the real root of x^3 + 4*x - 13.

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

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

Table of n, a(n) for n=1..105.

MacTutor History of Mathematics, Nicolo Tartaglia.

Index entries for algebraic numbers, degree 3

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 * A345384 A199742 A258112

Adjacent sequences: A257236 A257237 A257238 * A257240 A257241 A257242

Keyword

nonn,easy,cons

Author

Wolfdieter Lang, May 21 2015

Status

approved