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On Mordell's equation $ y\sp 2-k=x\sp 3$: a problem of Stolarsky

Author: Ray P. Steiner
Journal: Math. Comp. 46 (1986), 703-714
MSC: Primary 11D25; Secondary 11-04, 11Y50
MathSciNet review: 829640
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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{displaymath}\begin{array}{*{20}{c}} {x = 10,} \hfill & {y = \pm 1,} \hfil... ...l \\ {x = 174,} \hfill & {y = \pm 2295,} \hfill \\ \end{array} \end{displaymath}


$\displaystyle x = 22480,\quad y = \pm 3370501.$

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Keywords: Mordell's equation, Ellison's method, Davenport's lemma, linear forms in logarithms, Thue equations, cubic fields
Article copyright: © Copyright 1986 American Mathematical Society