An effective estimate for the sum of two cubes problem

Abstract

Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that $N_{f}(m)>\Omega\left((\log |m|)^{3 / 5}\right)$ when $f(x, y)=x^{3}+y^{3}$. In this paper, we obtain an explicit bound for $N_f(m)$, namely, showing that $N_{f}(m)>4.2\times 10^{-6}(\log |m|)^{11/13}$ (holds for infinitely many integers m), when $f(x, y)=x^{3}+y^{3}$.