A257236 Decimal expansion of the real root of 4*x^3 + 3*x - 40.

2, 0, 3, 8, 5, 1, 3, 1, 9, 8, 1, 2, 4, 5, 0, 6, 1, 7, 6, 8, 5, 7, 1, 3, 7, 4, 2, 3, 5, 4, 3, 1, 0, 2, 4, 8, 5, 1, 8, 5, 6, 2, 2, 1, 0, 9, 3, 0, 6, 2, 3, 9, 3, 4, 9, 9, 1, 0, 6, 8, 1, 4, 2, 7, 2, 1, 9, 6, 2, 5, 8, 9, 1, 1, 4, 2, 8, 1, 7, 5, 4, 9, 6, 2, 3, 5, 8, 7, 5, 1, 6, 8, 2, 9, 8, 4, 2

Offset

1,1

Comments

This is the solution to the second of thirty problems posed by Antonio Maria Fiore in the year 1535 to Niccolò Tartaglia. See the Alten et al. reference, p. 272.

See also the Fauvel and Gray reference, p. 254, where the problem is translated as "Find me two numbers in double proportion such that when the square of the larger number is multiplied by the smaller, and this product is added to the two original numbers, the result is forty, that is 40." The explanation given there is equivalent to the equation (2x)^2*x + x + 2x = 40, i.e., 4x^3 + 3x = 40.

References

H.-W. Alten et al., 4000 Jahre Algebra, 2. Auflage, Springer, 2014, p. 272.

John Fauvel and Jeremy Gray (eds.), The History of Mathematics: A Reader, Macmillan Press, The Open University, 1988.

Links

Table of n, a(n) for n=1..97.

MacTutor History of Mathematics, Nicolo Tartaglia.

Formula

The real solution of the equation 4*x^3 +3*x - 40 = 0 is x1 = (1/2)*((40 + sqrt(1601))^(1/3) - (-40 + sqrt(1601))^(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 = (1/2)*((40 + sqrt(1601))^(1/3) + (-40 + sqrt(1601))^(1/3)).

Example

x1 = 2.0385131981245061768571374235431...
y1 = 2.2705805554808669182617619994067...

Mathematica

RealDigits[N[Solve[4 x^3+3 x-40==0, x][[1, 1, 2]], 111]][[1]]
(* Vincenzo Librandi, May 09 2015 *)

Crossrefs

Cf. A257235, A257237.

Sequence in context: A334847 A131330 A022833 * A209259 A020826 A094346

Adjacent sequences: A257233 A257234 A257235 * A257237 A257238 A257239

Keyword

nonn,easy,cons

Author

Wolfdieter Lang, May 08 2015

Status

approved