%I #19 Jan 19 2019 04:15:43
%S 3,2,6,9,9,2,8,3,0,3,8,2,0,8,7,0,5,8,0,2,3,9,1,7,8,1,3,6,8,5,9,2,6,6,
%T 8,6,9,9,7,6,4,9,4,3,1,0,1,7,1,6,6,6,9,3,2,4,0,5,9,5,8,7,9,9,1,7,0,1,
%U 8,5,5,6,3,5,8,5,8,2,7,8,1,0,6,1,5,8,8,5,0,5,3,9,9,5,3,4,5,6,0,5
%N Decimal expansion of the positive member y of a triple (x, y, z) solving a certain historical system of three equations.
%C This number is the second member y of one of the two real triples (x, y, z) which solve the three equations i) x + y + z = 10, ii) x*z = y^2, iii) x^2 + y^2 = z^2. The corresponding numbers are x = 10*A248752 and z = A300071.
%C The other real solution has x = x2 = 10*A248750, y = y2 = -A300072 and z = z2 = A300073.
%C The two complex solutions have y3 = 5*(phi + sqrt(phi - 1)*i) with phi = A001622 and i = sqrt(-1), and x3 = y3 - (1/50)*(y3)^3, z3 = 10 - 2*y3 + (1/50)*y3^3.
%C The polynomial for the solutions Y = y/5 is P(Y) = Y^4 - 2*Y^3 - 2*Y^2 + 8*Y - 4, or in standard form p(U) = U^4 - (7/2)*U^2 + 5*U - 11/64, with U = Y - 1/2. This factorizes as p(U) = p1(U)*p2(U) with p1(U) = U^2 - (2*phi - 1)*U + 1/4 + phi and p2(U) = U^2 + (2*phi - 1)*U + 5/4 - phi.
%C This problem appears (see the Havil reference) in Abū Kāmil's Book on Algebra. Havil gives only the positive real solution (x, y, z) on p. 60.
%C Note that X = x/5, Y = y/5 and Z = z/5 solves i') X + Y + Z = 2, ii) X*Z = Y^2, iii) X^2 + Y^2 = Z^2.
%D Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58-60.
%H I.C. Karpinski, <a href="http://www.jstor.org/stable/2972073?seq=1#page_scan_tab_contents">The Algebra of Abu Kamil</a>, Amer. Math. Month. XXI,2 (1914), 37-48.
%H MacTutor History of Mathematics, <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Abu_Kamil.html">Abu Kamil Shuja</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ab%C5%AB_K%C4%81mil_Shuj%C4%81%CA%BF_ibn_Aslam">Abu Kamil</a>.
%F y = 5*(1 - phi + sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.
%e y = 3.26992830382087058023917813685926686997649431017166693240595879917018...
%e y/5 = 0.65398566076417411604783562737185337399529886203433338648119175983...
%t RealDigits[5 (1 - GoldenRatio + Sqrt[GoldenRatio]), 10, 100][[1]] (* _Bruno Berselli_, Mar 02 2018 *)
%Y Cf. A001622, A248750, A248752, A300071, A300072, A300073.
%K nonn,cons,easy
%O 1,1
%A _Wolfdieter Lang_, Mar 02 2018