Entry 13

In Question 441 of the Journal of the Indian Mathematical Society (JIMS), Ramanujan asked,

"Show that $(3x^2+5xy-5y^2)^3 +(4x^2-4xy+6y^2)^3 +(5x^2-5xy-3y^2)^3 =(6x^2-4xy+4y^2)^3$ and find other quadratic expressions satisfying similar relations."

There are in fact infinitely many such quadratic expressions. For general $a^3+b^3+c^3+d^3 = 0$ use the identity $$A^3+B^3+C^3+D^3 = (a^3+b^3+c^3+d^3)(x^2+wy^2)^3$$ and $A,B,C,D$ are quadratic forms,  $$\begin{align}A &= ax^2-v_1xy+bwy^2\\ B &= bx^2+v_1xy+awy^2\\ C &= cx^2+v_2xy+dwy^2\\ D &= dx^2-v_2xy+cwy^2\end{align}$$  where $\big(v_1,\, v_2,\, w\big) = \big(c^2-d^2,\; a^2-b^2,\; (a+b)(c+d)\big)$.