Equal sums of two cubes of binary quadratic forms

Abstract

We give a complete description of all solutions to the equation $f_1^3+f_2^3=f_3^3+f_4^3$ for quadratic forms $f_j\in \mathbb{C}[x,y]$ and show how Ramanujan's example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic $p\in \mathbb{C}[x,y]$ can be written as a sum of two cubes. The extreme example is $p(x,y)=xy(x^4-y^4)$, which has six such representations.

Submitted to International Journal of Number Theory, for the Special Issue on the Berndt Conference Proceedings