I searched numbers $n$ and $k$, such that the equation $$y^2=x^3+k$$ (Modell-curve-equation) has no solution modulo $n$ in order to find families of $k$'s such that $y^2=x^3+k$ has no integral solution and that this can be shown simply with modular arithmetic.
But for $2\le n\le 200$ and every $k$, I verified that always a solution in $\mathbb Z_n$ exists.
Is the Mordell-equation solveable in $\mathbb Z_n$ for every $k$ and every $n$ ?