A300070 Decimal expansion of the positive member y of a triple (x, y, z) solving a certain historical system of three equations.

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, 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, 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

Offset: 1

Author

Wolfdieter Lang, Mar 02 2018

Keywords

• nonn
• cons
• easy

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 other real solution has x = x2 = 10*A248750, y = y2 = -A300072 and z = z2 = A300073.

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.

Examples

			y = 3.26992830382087058023917813685926686997649431017166693240595879917018...
y/5 = 0.65398566076417411604783562737185337399529886203433338648119175983...
		

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.

Crossrefs

Cf. A001622, A248750, A248752, A300071, A300072, A300073.

Programs

Mathematica

RealDigits[5 (1 - GoldenRatio + Sqrt[GoldenRatio]), 10, 100][[1]] (* Bruno Berselli, Mar 02 2018 *)

Formula

y = 5*(1 - phi + sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.