An effective estimate for the sum of two cubes problem

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, $N_f(m)=\mathrm{\#}\{(x,y)\in {\mathbb{Z}}^2:f(x,y)=m\}$. In 1983, Silverman proved that $N_f(m)>\mathrm{\Omega }((\log \mathrm{\mid }m\mathrm{\mid }{)}^{3\mathrm{/}5})$ when $f(x,y)=x^3+y^3$. In this paper, we obtain an explicit bound for $N_f(m)$, namely, showing that $N_f(m)>4.2\times {10}^{-6}(\log \mathrm{\mid }m\mathrm{\mid }{)}^{11\mathrm{/}13}$ (holds for infinitely many integers m), when $f(x,y)=x^3+y^3$.