|
%I #20 Jan 19 2019 04:15:43
%S 9,4,5,0,2,6,8,1,9,1,3,1,9,8,1,9,0,6,2,2,8,5,0,4,6,4,8,0,5,1,5,6,4,8,
%T 0,4,7,1,7,9,5,8,6,1,0,8,2,2,9,2,9,5,5,5,3,7,6,0,4,4,5,0,2,6,2,2,2,7,
%U 9,0,1,9,1,7,7,4,8,5,2,3,0,7,6,8,7,9,5,7,0,9,5,8,8,9,2,5,6,9,8
%N Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations.
%C The system of three equations is
%C x + y + z = 10,
%C x*z = y^2,
%C x^2 + y^2 = z^2.
%C See A300070 for the Havil reference and links to Abū Kāmil who considered this system. This real solution was not given in Havils's book.
%C This solution is x = x2:= 10*A248750, -y=-y2= present entry, z = z2 = A300073.
%C The other real solution with positive y is x = 10*A248752, y = A300070, z = A300071.
%C Note that X2 = x2/5, -Y2 = -y2/5 and Z2 = z2/5 solve the system of equations i') X2 + Y2 + Z2 = 2, ii) X2*Z2 = (Y2)^2 and iii) (X2)^2 + (Y2)^2 = (Z2)^2.
%F -y2 = 5*(1- phi - sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.
%e -y2 = 9.450268191319819062285046480515648047179586108229295553760445026222...
%e -y2/5 = 1.8900536382639638124570092961031296094359172216458591107520890052...
%t RealDigits[5 (1 - GoldenRatio - Sqrt[GoldenRatio]), 10, 100][[1]] (* _Bruno Berselli_, Mar 02 2018 *)
%Y Cf. A001622, A248750, A248752, A300070, A300071, A300073.
%K nonn,cons,easy
%O 1,1
%A _Wolfdieter Lang_, Mar 02 2018
|
