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A257237 Decimal expansion of the real root of x^3 + x - 500. 4
7, 8, 9, 5, 0, 0, 8, 2, 8, 5, 5, 3, 5, 9, 1, 1, 4, 7, 8, 0, 4, 8, 9, 1, 1, 6, 0, 6, 3, 9, 5, 7, 3, 1, 0, 7, 1, 2, 4, 7, 2, 8, 7, 8, 5, 5, 5, 5, 0, 6, 9, 8, 0, 9, 6, 2, 3, 8, 9, 8, 5, 6, 0, 0, 4, 3, 7, 6, 4, 4, 0, 4, 6, 0, 6, 3, 0, 9, 4, 2, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

See the Fauvel and Gray reference, p. 254, where this problem is translated as "A man sells a sapphire for 500 ducats, making a profit of the cube root of his capital. How much is this profit?" and the explanation given there is [x^3 + x = 500.] One assumes that the capital is the value of the sapphire before the selling. Note that the authors use 'profit of the cube root', whereas in the Alten et al. reference this is translated as 'Gewinn in der dritten Potenz'. But here the 'cube root' interpretation seems more plausible because then from c + c^(1/3) = 500 the capital c turns out to be  500 - x1, about 492.10 ducats, and the profit c^(1/3) is about 7.90 ducats, that is about 1.6 percent. (Not a big deal, though.) The Alten et al. version would give for the capital x1 about 7.10 ducats and a huge profit of about 492.10 ducats. One would like to read the original.

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..81.

MacTutor History of Mathematics, Nicolo Tartaglia.

FORMULA

The real solution of the equation x^3 + x - 500 = 0 is x1 = (1/3)*((6750 + 3*sqrt(5062503))^(1/3) - (-6750 + 3*sqrt(5062503))^(1/3)).

  6750 = 2*(3*5)^3, 5062503 = 3*229*7369.

The two complex solutions are a + i*b and a - i*b with a = -x1/2 and b = i*sqrt(3)*y1/2, where y1 =

(1/3)*((6750 + 3*sqrt(5062503))^(1/3) + (-6750   + 3*sqrt(5062503))^(1/3)).

EXAMPLE

x1 = 7.895008285535911478048911606395731...

y1 = 7.979003018047682483096396690816339...

MATHEMATICA

RealDigits[N[Solve[x^3 + x - 500==0, x][[1]][[1, 2]], 120]][[1]] (* Vincenzo Librandi, May 09 2015 *)

PROG

(PARI) polrootsreal(x^3 + x - 500)[1] \\ Charles R Greathouse IV, May 11 2015

CROSSREFS

Cf. A257235, A257236.

Sequence in context: A284363 A114514 A011471 * A242022 A085676 A036793

Adjacent sequences:  A257234 A257235 A257236 * A257238 A257239 A257240

KEYWORD

nonn,easy,cons

AUTHOR

Wolfdieter Lang, May 08 2015

STATUS

approved

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Last modified September 19 03:31 EDT 2021. Contains 347550 sequences. (Running on oeis4.)