A300072 Decimal expansion of the positive member -y of a triple (x, y, z) solving a certain historical system of three equations.

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, 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, 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

Offset: 1

Author

Wolfdieter Lang, Mar 02 2018

Keywords

• nonn
• cons
• easy

Comments

The system of three equations is

x + y + z = 10,

x*z = y^2,

x^2 + y^2 = z^2.

See A300070 for the Havil reference and links to Abū Kāmil, who considered this system. This real solution was not given in Havil's book.

This solution is x = x2 := 10*A248750, -y = -y2 = present entry, z = z2 = A300073.

The other real solution with positive y is x = 10*A248752, y = A300070, z = A300071.

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.

Examples

			-y2 = 9.450268191319819062285046480515648047179586108229295553760445026222...
-y2/5 = 1.8900536382639638124570092961031296094359172216458591107520890052...
		

Crossrefs

Cf. A001622, A248750, A248752, A300070, A300071, A300073.

Programs

Mathematica

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

Formula

-y2 = 5*(1 - phi - sqrt(phi)), with the golden section phi = (1 + sqrt(5))/2 = A001622.

The minimal polynomial is x^4 + 10*x^3 - 50*x^2 - 1000*x - 2500. - Joerg Arndt, Jul 21 2025