On Mordell’s equation $y^ 2-k=x^ 3$: a problem of Stolarsky
HTML articles powered by AMS MathViewer
- by Ray P. Steiner PDF
- Math. Comp. 46 (1986), 703-714 Request permission
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 \[ x = 22480,\quad y = \pm 3370501.\]References
- V. I. Baulin, On an indeterminate equation of the third degree with least positive discriminant, Tul′sk. Gos. Ped. Inst. Učen. Zap. Fiz.-Mat. Nauk Vyp. 7 (1960), 138–170 (Russian). MR 0199149
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
- B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744 W. J. Ellison, "Recipes for solving Diophantine problems by Baker’s method," Publ. Mathématiques, v. Ann. 1, Fasc. 1, 1972.
- W. J. Ellison, F. Ellison, J. Pesek, C. E. Stahl, and D. S. Stall, The Diophantine equation $y^{2}+k=x^{3}$, J. Number Theory 4 (1972), 107–117. MR 316376, DOI 10.1016/0022-314X(72)90058-3
- Ove Hemer, On the solvability of the Diophantine equation $ax^2+by^2+cz^2=0$ in imaginary Euclidean quadratic fields, Ark. Mat. 2 (1952), 57–82. MR 49917, DOI 10.1007/BF02591382
- Wilhelm Ljunggren, Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante, Acta Math. 75 (1943), 1–21 (German). MR 17303, DOI 10.1007/BF02404100
- A. Pethő, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen 30 (1983), no. 1-2, 117–127. MR 733078
- V. G. Sprindzhuk, Klassicheskie diofantovy uravneniya ot dvukh neizvestnykh, “Nauka”, Moscow, 1982 (Russian). MR 685430
- Kenneth B. Stolarsky, Algebraic numbers and Diophantine approximation, Pure and Applied Mathematics, No. 26, Marcel Dekker, Inc., New York, 1974. MR 0374041
- Nicholas Tzanakis, The Diophantine equation $x^{3}-3xy^{2}-y^{3}=1$ and related equations, J. Number Theory 18 (1984), no. 2, 192–205. MR 741950, DOI 10.1016/0022-314X(84)90053-2
- Michel Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257–283. MR 598881, DOI 10.4064/aa-37-1-257-283
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 46 (1986), 703-714
- MSC: Primary 11D25; Secondary 11-04, 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-1986-0829640-3
- MathSciNet review: 829640