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. Use the identity (by yours truly) $$(ax^2-v_1xy+bwy^2)^3 + (bx^2+v_1xy+awy^2)^3 + (cx^2+v_2xy+dwy^2)^3 + (dx^2-v_2xy+cwy^2)^3 \\= (a^3+b^3+c^3+d^3)(x^2+wy^2)^3$$ where $v_1=c^2-d^2,\; v_2=a^2-b^2,\,$ and $w=(a+b)(c+d)$. Thus all we need is an initial solution to $a^3+b^3+c^3+d^3=0$ and the identity guarantees an infinite more.