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 $N_f(m)>\hat{I}\text{c◯}\left((\log |m|{)}^{3/5}\right)$ 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 $f(x,y)=x^3+y^3$.

7 pages, some calculations corrected, and the results of Prof. Stewart added