On Elliptic Diophantine Equations That Defy Thue's Method: The Case of the Ochoa Curve

REFERENCES

• Batut , C. , Bernardi , D. , Cohen , H. and Olivier , M. User's Guide to Pari-GP. [Batut et al. 1992], This manual is part of the program distribution, available by anonymous ftp from the host pari@ceremab. u-bordeaux.fr
   Google Scholar
• Bremner , A. 1989 . “On the equation Y2= XX2+p)”. ” . In Number Theory and Applications Edited by: Mollin , R. A. 3 – 23 . Dordrecht : Kluwer. . [Bremner 1989]
   Google Scholar
• Bremner , A. and Cassels , J. W. S. 1984 . “On the equation Y2= XX2+ p)” . Math. Comp. , 42 : 257 – 264 . [Bremner and Cassels 1984]
   Web of Science ®Google Scholar
• Cassels , J. W. S. 1978 . Rational Quadratic Forms London and New York : Academic Press. . [Cassels 1978], London Math. Soc. Monographs 13
   Google Scholar
• Cassels , J. W. S. 1991 . Lectures on Elliptic Curves Cambridge and New York : Cambridge Univ. Press. . [Cassels 1991], London Math. Soc. Student Texts 24
   Google Scholar
• Char , B. W. 1991 . Maple V Language Reference Manual and Maple V Library Reference Manual New York : Springer. . [Char et al. 1991]
   Google Scholar
• Connell , I. “Apecs (Arithmetic of Plane Elliptic Curves)” [Connell 1994], This large and very useful collection of Maple procedures can be obtained by anonymous ftp from the host math.mcgill.ca
   Google Scholar
• Cremona , J. E. 1992 . Algorithms for Modular Elliptic Curves Cambridge and New York : Cambridge Univ. Press. . [Cremona 1992]
   Google Scholar
• David , S. 1991–1992 . “Minorations de formes linéaires de logarithmes elliptiques” . Publications mathématiques de I'Université Pierre et Marie Curie , 106 [David 1992], Problèmes diophantiens, exposé no. 3
   Google Scholar
• Gebel , J. , Pethö , A. and Zimmer , H. G. “Computing integral points on elliptic curves” . Acta Arithmetica. , [Gebel et al.], to appear
   Web of Science ®Google Scholar
• Guy , R. K. 1990 . “The Ochoa curve” . Crux Mathematicorum , 16 ( 3 ) : 65 – 69 . [Guy 1990]
   Google Scholar
• Knapp , A. W. 1992 . Elliptic Curves Princeton : Princeton Univ. Press. . [Knapp 1992], Mathematical Notes 40
   Google Scholar
• Mestre , J.-F. 1986 . “Formules explicites et minorations de conducteurs de variétés algébriques” . Compositio Mathematica , 58 : 209 – 232 . [Mestre 1986]
   Google Scholar
• Silverman , J. H. 1986 . The Arithmetic of Elliptic Curves New York : Springer. . [Silverman 1986], Graduate Texts in Mathematics 106
   Google Scholar
• Stroeker , R. J. 1984 . “How to solve a diophantine equation” . Amer. Math. Monthly , 91 : 385 – 392 . [Stroeker 1984]
   Web of Science ®Google Scholar
• Stroeker , R. J. and Top , J. 1994 . “On the equation Y²= (X + p)(X²+p²)” [Stroeker and Top 1994], to appear in Rocky Mountain J. Math.
   Google Scholar
• Stroeker , R. J. and Tzanakis , N. 1988 . “On the application of Skolem's p-adic method to the solution of Thue equations” . J. Number Theory , 29 : 166 – 195 . [Stroeker and Tzanakis 1988]
   Web of Science ®Google Scholar
• Stroeker , R. J. and Tzanakis , N. 1994 . “Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms” . Acta Arithmetica , 67 ( 2 ) : 177 – 196 . [Stroeker and Tzanakis 1994]
   Web of Science ®Google Scholar
• Tzanakis , N. and de Weger , B. M. M. 1989 . “On the practical solution of the Thue equation” . J. Number Theory , 31 ( 2 ) : 99 – 132 . [Tzanakis and de Weger 1989]
   Web of Science ®Google Scholar
• de Weger , B. M. M. 1989 . “Algorithms for Diophantine equations” Amsterdam : Stichting Mathematisch Centrum. . [de Weger 1989], CWI Tract 65
   Google Scholar
• de Weger , B. M. M. “Solving Elliptic Diophantine Equations by Bilu's Method” 17 Econometric Institute, Erasmus University Rotterdam. . [de Weger 1994], Report 9469/B
   Google Scholar
• Zagier , D. 1987 . “Large integral points on elliptic curves” . Math. Comp. , 48 : 425 – 436 . [Zagier 1987]
   Web of Science ®Google Scholar