OFFSET
1,1
COMMENTS
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.
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.
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.
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.
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.
REFERENCES
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58-60.
LINKS
I.C. Karpinski, The Algebra of Abu Kamil, Amer. Math. Month. XXI,2 (1914), 37-48.
MacTutor History of Mathematics, Abu Kamil Shuja.
Wikipedia, Abu Kamil.
FORMULA
y = 5*(1 - phi + sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.
EXAMPLE
y = 3.26992830382087058023917813685926686997649431017166693240595879917018...
y/5 = 0.65398566076417411604783562737185337399529886203433338648119175983...
MATHEMATICA
RealDigits[5 (1 - GoldenRatio + Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Mar 02 2018
STATUS
approved