A lower bound for the number of integral solutions of Mordell equation

Abstract

For a nonzero integer d, a celebrated Siegel Theorem says that the number N(d) of integral solutions of Mordell equation y² + x³ = d is finite. We find a lower bound for N(d), showing that the number of solutions of Mordell equation increases dramatically. We also prove that for any positive integer n, there is an integer square multiply represented by Mordell equations, i.e., k² = y₁² + x₁³ = y₂² + x₂³ = ··· = y_n² + x_n³.