cp's OEIS Frontend

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.

See A300071 for the system of equations and references and links.

The current triple is (x = 10*A248752, A300070, z = this entry).

The other real solution is (x = x2 = 10*A248750, y = y2 = A300072, z = z2 = A300073).

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.