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Mordell’s equation: a classical approach

Part of: Diophantine equations Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 September 2015

Michael A. Bennett  and
Amir Ghadermarzi
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2 email [email protected]
Amir Ghadermarzi
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2 email [email protected]

Abstract

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We solve the Diophantine equation $Y^{2}=X^{3}+k$ for all nonzero integers $k$ with $|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

MSC classification

Primary: 11D25: Cubic and quartic equations 11G05: Elliptic curves over global fields
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 633 - 646
Copyright
© The Author(s) 2015 

References

Baker, A., ‘On the representation of integers by binary forms’, Philos. Trans. A 263 (1968) 173–208.Google Scholar
Belabas, K., ‘A fast algorithm to compute cubic fields’, Math. Comp. 66 (1997) 1213–1237.Google Scholar
Belabas, K. and Cohen, H., ‘Binary cubic forms and cubic number fields’, Organic mathematics (Burnaby, BC, 1995) , CMS Conference Proceedings 20 (American Mathematical Society, Providence, RI, 1997) 175–204.Google Scholar
Berwick, W. E. H. and Mathews, G. B., ‘On the reduction of arithmetical binary cubic forms which have a negative determinant’, Proc. Lond. Math. Soc. (2) 10 (1911) 43–53.Google Scholar
Cremona, J., ‘Reduction of binary cubic and quartic forms’, LMS J. Comput. Math. 4 (1999) 64–94.Google Scholar
David, S. and Hirata-Kohno, N., ‘Linear forms in elliptic logarithms’, J. reine angew. Math. 628 (2009) 37–89.Google Scholar
Davenport, H., ‘The reduction of a binary cubic form. I’, J. Lond. Math. Soc. 20 (1945) 14–22.Google Scholar
Davenport, H., ‘The reduction of a binary cubic form. II’, J. Lond. Math. Soc. 20 (1945) 139–147.Google Scholar
Davenport, H. and Heilbronn, H., ‘On the density of discriminants of cubic fields. II’, Proc. R. Soc. Lond. A 322 (1971) 405–420.Google Scholar
Delone, B. N. and Faddeev, D. K., The theory of irrationalities of the third degree , Translations of Mathematical Monographs 10 (American Mathematical Society, Providence, RI, 1964).Google Scholar
Dickson, L. E., History of the theory of numbers, Vol 2: Diophantine analysis (Carnegie Institute, Washington DC, 1920; reprinted Chelsea, New York, 1952).Google Scholar
Elkies, N., ‘Rational points near curves and small nonzero |x 3 − y 2| via lattice reduction’, Algorithmic number theory (Proceedings of ANTS-IV) , Lecture Notes in Computer Science 1838 (ed. Bosma, W.; Springer, Berlin, 2000) 33–63.Google Scholar
Ellison, W. J., Ellison, F., Pesek, J., Stahl, C. E. and Stall, D. S., ‘The Diophantine equation y 2 + k = x 3 ’, J. Number Theory 4 (1972) 107–117.Google Scholar
Fieker, C., Gaál, I. and Pohst, M., ‘On computing integral points of a Mordell curve over rational function fields in characteristic > 3’, J. Number Theory 133 (2013) 738–750.CrossRefGoogle Scholar
Finkelstein, R. and London, H., ‘On Mordell’s equation y 2 − k = x 3 : an interesting case of Sierpinski’, J. Number Theory 2 (1970) 310–321.Google Scholar
Gebel, J., Pethő, A. and Zimmer, H. G., ‘Computing integral points on elliptic curves’, Acta Arith. 68 (1994) no. 2, 171–192.Google Scholar
Gebel, J., Pethő, A. and Zimmer, H. G., ‘On Mordell’s equations’, Compos. Math. 110 (1998) no. 3, 335–367.Google Scholar
Gross, R. and Silverman, J., ‘ S-integer points on elliptic curves’, Pacific J. Math. 167 (1995) 263–288.Google Scholar
Hall, M., ‘The Diophantine equation x 3 − y 2 = k ’, Computers in number theory (eds Atkin, A. and Birch, B.; Academic Press, London, 1971) 173–198.Google Scholar
Hemer, O., ‘Notes on the Diophantine equation y 2 − k = x 3 ’, Ark. Mat. 3 (1954) 67–77.Google Scholar
Hermite, C., ‘Note sur la réduction des formes homogènes à coefficients entiers et à deux indétermineées’, J. reine angew. Math. 36 (1848) 357–364.Google Scholar
Hermite, C., ‘Sur la réduction des formes cubiques à deux indétermineées’, C. R. Math. Acad. Sci. Paris 48 (1859) 351–357.Google Scholar
Ingram, P., ‘Multiples of integral points on elliptic curves’, J. Number Theory 129 (2009) no. 1, 182–208.Google Scholar
Jätzschmann, A., Zur Bestimmung ganzer Punkte auf elliptischen Kurven (Diplomarbeit, Technische Universität, Berlin, 2010).Google Scholar
Jiménez Calvo, I., Herranz, J. and Sáez, G., ‘A new algorithm to search for small nonzero |x 3 − y 2| values’, Math. Comp. 78 (2009) 2435–2444.Google Scholar
Julia, G., ‘Étude sur les formes binaires non quadratiques à indéterminées rélles ou complexes’, Mem. Acad. Sci. l’Inst. France 55 (1917) 1–293.Google Scholar
Lang, S., Elliptic curves: Diophantine analysis , Grundlehren der Mathematischen Wissenschaften 231 (Springer, Berlin, 1978).Google Scholar
Ljunggren, W., ‘The diophantine equation y 2 = x 3 − k ’, Acta Arith. 8 (1961) 451–465.CrossRefGoogle Scholar
Mordell, L. J., ‘The diophantine equation y 2 − k = x 3 ’, Proc. Lond. Math. Soc. (2) 13 (1913) 60–80.Google Scholar
Mordell, L. J., ‘Indeterminate equations of the third and fourth degree’, Quart. J. Pure Appl. Math. 45 (1914) 170–186.Google Scholar
Mordell, L. J., ‘A statement of Fermat’, Proc. Lond. Math. Soc. (2) 18 (1919) v–vi.Google Scholar
Mordell, L. J., Diophantine equations (Academic Press, London, 1969).Google Scholar
Smart, N. P., ‘S-integral points on elliptic curves’, Math. Proc. Cambridge Philos. Soc. 116 (1994).CrossRefGoogle Scholar
Stroeker, R. J. and Tzanakis, N., ‘Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms’, Acta Arith. 67 (1994) 177–196.Google Scholar
Thue, A., ‘Über Annäherungswerte algebraischer Zahlen’, J. reine angew Math. 135 (1909) 284–305.CrossRefGoogle Scholar
Tzanakis, N. and de Weger, B. M. M., ‘On the practical solutions of the Thue equation’, J. Number Theory 31 (1989) 99–132.Google Scholar
Wildanger, K., ‘Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern mit einer Anwendung auf die Bestimmung aller ganzen Punkte einer Mordellschen Kurve’, Thesis, Technical University, Berlin, 1997.Google Scholar
Zagier, D., ‘Large integral points on elliptic curves’, Math. Comp. 48 (1987) no. 177, 425–436.Google Scholar