A parametric family of quintic Thue equations

Authors:
István Gaál and Günter Lettl

Journal:
Math. Comp. **69** (2000), 851-859

MSC (1991):
Primary 11D57; Secondary 11Y50

DOI:
https://doi.org/10.1090/S0025-5718-99-01155-2

Published electronically:
May 24, 1999

MathSciNet review:
1659855

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For an integral parameter we investigate the family of Thue equations

originating from Emma Lehmer's family of quintic fields, and show that for the only solutions are the trivial ones with or . Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

**1.**A. Baker & G. Wüstholz,*Logarithmic forms and group varieties*, J. Reine Angew. Math.**442**(1993), 19-62. MR**94i:11050****2.**H. Darmon,*Note on a polynomial of Emma Lehmer*, Math. Comp.**56**(1991), 795-800. MR**91i:11149****3.**I. Gaál & M. Pohst,*Power integral bases in a parametric family of totally real cyclic quintics*, Math. Comp.**66**(1997), 1689-1696. MR**98a:11160****4.**C. Heuberger,*On a family of quintic Thue equations*, J. Symbolic Comput.**26**(1998), 173-185. CMP**98:16****5.**C. Heuberger, A. Peth\H{o} & R.F. Tichy,*Complete solution of parametrized Thue equations*, Acta Math. Inform. Univ. Ostraviensis**6**(1998), 93-114.**6.**E. Lehmer,*Connection between Gaussian periods and cyclic units*, Math. Comp.**50**(1988), 535-541. MR**89h:11067a****7.**G. Lettl & A. Peth\H{o},*Complete solution of a family of quartic Thue equations*, Abh. Math. Sem. Univ. Hamburg**65**(1995), 365-383. MR**96h:11019****8.**M. Mignotte, A. Peth\H{o} & R. Roth,*Complete solutions of a family of quartic Thue and index form equations*, Math. Comp.**65**(1996), 341-354. MR**96d:11034****9.**A. Peth\H{o},*Complete solutions to families of quartic Thue equations*, Math. Comp.**57**(1991), 777-798. MR**92e:11023****10.**A. Peth\H{o} & R.F. Tichy,*On two-parametric quartic families of Diophantine problems*, J. Symbolic Comput.**26**(1998), 151-171. CMP**98:16****11.**R. Schoof & L. Washington,*Quintic polynomials and real cyclotomic fields with large class numbers*, Math. Comp.**50**(1988), 543-556. MR**89h:11067b****12.**E. Thomas,*Complete solutions to a family of cubic diophantine equations*, J. Number Theory**34**(1990), 235-250. MR**91b:11027****13.**E. Thomas,*Solutions to certain families of Thue equations*, J. Number Theory**43**(1993), 319-369. MR**94b:11028**

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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

DOI:
https://doi.org/10.1090/S0025-5718-99-01155-2

Keywords:
Parametric Thue equation,
Baker's method

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

Article copyright:
© Copyright 2000
American Mathematical Society