

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
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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. 1316.


LINKS

Table of n, a(n) for n=1..129.
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^33*x10)[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



