A parametric family of quintic Thue equations
HTML articles powered by AMS MathViewer
- by István Gaál and Günter Lettl PDF
- Math. Comp. 69 (2000), 851-859 Request permission
Abstract:
For an integral parameter $t \in \mathbb {Z}$ we investigate the family of Thue equations \begin{multline*} F(x,y) = x^{5} + (t-1)^{2}x^{4}y - (2t^{3}+4t+4)x^{3}y^{2}\ + (t^{4}+t^{3}+2t^{2}+4t-3)x^{2}y^{3} + (t^{3}+t^{2}+5t+3)xy^{4} + y^{5} = \pm 1 , \end{multline*} originating from Emma Lehmer’s family of quintic fields, and show that for $|t| \ge 3.28 \cdot 10^{15}$ the only solutions are the trivial ones with $x=0$ or $y=0$. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.References
- A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62. MR 1234835, DOI 10.1515/crll.1993.442.19
- Henri Darmon, Note on a polynomial of Emma Lehmer, Math. Comp. 56 (1991), no. 194, 795–800. MR 1068821, DOI 10.1090/S0025-5718-1991-1068821-5
- István Gaál and Michael Pohst, Power integral bases in a parametric family of totally real cyclic quintics, Math. Comp. 66 (1997), no. 220, 1689–1696. MR 1423074, DOI 10.1090/S0025-5718-97-00868-5
- C. Heuberger, On a family of quintic Thue equations, J. Symbolic Comput. 26 (1998), 173–185.
- C. Heuberger, A. Pethő & R.F. Tichy, Complete solution of parametrized Thue equations, Acta Math. Inform. Univ. Ostraviensis 6 (1998), 93–114.
- Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535–541. MR 929551, DOI 10.1090/S0025-5718-1988-0929551-0
- G. Lettl and A. Pethő, Complete solution of a family of quartic Thue equations, Abh. Math. Sem. Univ. Hamburg 65 (1995), 365–383. MR 1359142, DOI 10.1007/BF02953340
- Maurice Mignotte, Attila Pethő, and Ralf Roth, Complete solutions of a family of quartic Thue and index form equations, Math. Comp. 65 (1996), no. 213, 341–354. MR 1316596, DOI 10.1090/S0025-5718-96-00662-X
- Attila Pethő, Complete solutions to families of quartic Thue equations, Math. Comp. 57 (1991), no. 196, 777–798. MR 1094956, DOI 10.1090/S0025-5718-1991-1094956-7
- A. Pethő & R.F. Tichy, On two-parametric quartic families of Diophantine problems, J. Symbolic Comput. 26 (1998), 151–171.
- Emma Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), no. 182, 535–541. MR 929551, DOI 10.1090/S0025-5718-1988-0929551-0
- Emery Thomas, Complete solutions to a family of cubic Diophantine equations, J. Number Theory 34 (1990), no. 2, 235–250. MR 1042497, DOI 10.1016/0022-314X(90)90154-J
- Emery Thomas, Solutions to certain families of Thue equations, J. Number Theory 43 (1993), no. 3, 319–369. MR 1212687, DOI 10.1006/jnth.1993.1024
Additional Information
- István Gaál
- Affiliation: Kossuth Lajos University, Mathematical Institute, H–4010 Debrecen Pf.12., Hungary
- Email: igaal@math.klte.hu
- Günter Lettl
- Affiliation: Karl-Franzens-Universität Graz, Institut für Mathematik, A–8010 Graz, Heinrichstraße 36, Austria
- Email: guenter.lettl@kfunigraz.ac.at
- Received by editor(s): December 12, 1997
- Received by editor(s) in revised form: July 14, 1998
- Published electronically: May 24, 1999
- Additional Notes: The first author’s research was supported in part by Grants 16791 and 16975 from the Hungarian National Foundation for Scientific Research
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 851-859
- MSC (1991): Primary 11D57; Secondary 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-99-01155-2
- MathSciNet review: 1659855