On Mordell’s equation $y^ 2-k=x^ 3$: a problem of Stolarsky

Abstract:

On page 1 of his book Algebraic Numbers and Diophantine Approximation, K. B. Stolarsky posed the problem of solving the equation ${y^2} + 999 = {x^3}$ in positive integers. In the present paper we refine some techniques of Ellison and Pethö and show that the complete set of integer solutions of Stolarsky’s equation is \[ \begin {array}{*{20}{c}} {x = 10,} \hfill & {y = \pm 1,} \hfill \\ {x = 12,} \hfill & {y = \pm 27,} \hfill \\ {x = 40,} \hfill & {y = \pm 251,} \hfill \\ {x = 147,} \hfill & {y = \pm 1782,} \hfill \\ {x = 174,} \hfill & {y = \pm 2295,} \hfill \\ \end {array} \] and \[ x = 22480,\quad y = \pm 3370501.\]