On Mordell’s equation 𝑦²-𝑘=𝑥³: 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 {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 \[ x = 10 , a m p ; y = ± 1 , x = 12 , a m p ; y = ± 27 , x = 40 , a m p ; y = ± 251 , x = 147 , a m p ; y = ± 1782 , x = 174 , a m p ; y = ± 2295 , \begin {array}{*{20}{c}} {x = 10,} \hfill & {y = \pm 1,} \hfill \\ {x = 12,} \hfill & {y = \pm 27,} \hfill \\ {x = 40,} \hfill & {y = \pm 251,} \hfill \\ {x = 147,} \hfill & {y = \pm 1782,} \hfill \\ {x = 174,} \hfill & {y = \pm 2295,} \hfill \\ \end {array} \] and \[ x = 22480 , y = ± 3370501. x = 22480,\quad y = \pm 3370501. \]