An Effective Lower Bound on the Integer Cube Sum

Let f(x, y) ∈ Z[x, y] be a cubic form with non-zero discriminant, and for each integer m ∈ Z, let, Nf(m)=#\(x, y) ∈ Z²: f(x, y)=m\. In 1983, Silverman proved that Nf(m)>Ω((log |m|)3 / 5) when f(x, y)=x³+y³. In this paper, we obtain an explicit bound for Nf(m), namely, showing that Nf(m)>13× 10⁻⁶(log |m|)11/13 (holds for infinitely many integers m), when f(x, y)=x³+y³.