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

1,1

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 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.

References

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58-60.

Links

Table of n, a(n) for n=1..100.

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

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

Sequence in context: A210738 A210601 A197493 * A171632 A245609 A365789

Adjacent sequences: A300067 A300068 A300069 * A300071 A300072 A300073

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Mar 02 2018

Status

approved