A257236 - OEIS

%I #25 Sep 20 2015 08:25:07

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

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

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

%H MacTutor History of Mathematics, <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Tartaglia.html">Nicolo Tartaglia</a>.

%F 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)).

%F 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)).

%e x1 = 2.0385131981245061768571374235431...

%e y1 = 2.2705805554808669182617619994067...

%t RealDigits[N[Solve[4 x^3+3 x-40==0,x][[1,1,2]],111]][[1]]

%t (* _Vincenzo Librandi_, May 09 2015 *)

%Y Cf. A257235, A257237.

%K nonn,easy,cons

%O 1,1

%A _Wolfdieter Lang_, May 08 2015