Simple families of Thue inequalities

Authors:
Günter Lettl, Attila Petho and Paul Voutier

Journal:
Trans. Amer. Math. Soc. **351** (1999), 1871-1894

MSC (1991):
Primary 11J25, 11J82; Secondary 11D25, 11D41

DOI:
https://doi.org/10.1090/S0002-9947-99-02244-8

Published electronically:
January 26, 1999

MathSciNet review:
1487624

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.

**[1]**A. Baker,*Rational approximations to and other algebraic numbers*, Quart. J. Math. Oxford**15**(1964), 375-383. MR**30:1977****[2]**M. Bennett,*Effective measures of irrationality for certain algebraic numbers*, J. Austral. Math. Soc. Ser. A**62**(1997), 329-344. MR**98c:11070****[3]**Chen Jian Hua,*A new solution of the Diophantine equation*, J. Number Theory**48**(1994), 62-74. MR**95i:11019****[4]**Chen Jian Hua and P. M. Voutier,*Complete solution of the Diophantine equation and a related family of quartic Thue equations*, J. Number Theory**62**(1997), 71-99. MR**97m:11039****[5]**G. V. Chudnovsky,*On the method of Thue-Siegel*, Ann. of Math.**117**(1983), 325-382. MR**85g:11058****[6]**J. H. E. Cohn,*Equations with equivalent roots*, Acta Arith.**34**(1977), 37-41. MR**56:15554****[7]**D. Easton,*Effective irrationality measures for certain algebraic numbers*, Math. Comp.**46**(1986), 613-622. MR**87f:11047****[8]**M. N. Gras,*Familles d'unités dans les extensions cycliques réelles de degré de*, Publ. Math. Fac. Sci. Besançon (1984 - 1986), fasc. 2, 27 pp. MR**88k:11078****[9]**A. J. Lazarus,*On the class number and unit index of simplest quartic fields*, Nagoya Math. J.**121**(1991), 1-13. MR**92a:11129****[10]**G. Lettl and 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****[11]**G. Lettl, A. Peth\H{o} and P. Voutier,*On the arithmetic of simplest sextic fields and related Thue equations*, Number Theory: Diophantine, Computational and Algebraic Aspects (K. Gy\H{o}ry, A. Peth\H{o} and V.T. Sós, eds.), Walter de Gruyter Publ. Co., 1998, 331-348.**[12]**F. Lorenz,*Lineare Algebra II*, BI-Wiss. Verlag Mannheim, 1989. MR**90f:15002****[13]**K. S. McCurley,*Explicit estimates for and*, Math. Comp.**42**(1984), 265-286. MR**85g:11085****[14]**M. Mignotte,*Verification of a conjecture of E. Thomas*, J. Number Theory**44**(1993), 172-177. MR**94m:11035****[15]**M. Mignotte, A. Peth\H{o}, F. Lemmermeyer,*On the family of Thue equations*, Acta Arith.**76**(1996), 245-269. MR**97k:11039****[16]**M. Mignotte, A. Peth\H{o}, R. Roth,*Complete solutions of quartic Thue and index form equations*, Math. Comp.**65**(1996), 341-354. MR**96d:11034****[17]**A. Peth\H{o},*On the resolution of Thue inequalities*, J. Symbolic Comput.**4**(1987), 103-109. MR**89b:11030****[18]**-,*Complete solutions to families of quartic Thue equations*, Math. Comp.**57**(1991), 777-798. MR**92e:11023****[19]**O. Ramaré and R. Rumely,*Primes in arithmetic progressions*, Math. Comp.**65**(1996), 397-425. MR**97a:11144****[20]**J. B. Rosser and L. Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64-94. MR**25:1139****[21]**R. Schoof and L. C. Washington,*Quintic polynomials and real cyclotomic fields with large class numbers*, Math. Comp.**50**(1988), 543-556. MR**89h:11067b****[22]**D. Shanks,*The simplest cubic fields*, Math. Comp.**28**(1974), 1134-1152. MR**50:4537****[23]**E. Thomas,*Complete solutions to a family of cubic diophantine equations*, J. Number Theory**34**(1990), 235-250. MR**91b:11027****[24]**P. M. Voutier,*Rational approximations to and other algebraic numbers revisited*, Indag. Math. (to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
11J25,
11J82,
11D25,
11D41

Retrieve articles in all journals with MSC (1991): 11J25, 11J82, 11D25, 11D41

Additional Information

**Günter Lettl**

Affiliation:
Institut für Mathematik, Karl-Franzens-Universität, Heinrichstraße 36, A-8010 Graz, Austria

Email:
guenter.lettl@kfunigraz.ac.at

**Attila Petho**

Affiliation:
Department of Mathematics and Informatics, Lajos Kossuth University, P.O. Box 12, H-4010 Debrecen, Hungary

Email:
pethoe@math.klte.hu

**Paul Voutier**

Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado 80309

Address at time of publication:
Optrak Distribution Software Ltd., Cawthorne House, 51 St. Andrew Street, Hertford SG14 1HZ, Great Britain

Email:
paul@optrak.co.uk

DOI:
https://doi.org/10.1090/S0002-9947-99-02244-8

Received by editor(s):
March 31, 1997

Published electronically:
January 26, 1999

Additional Notes:
Research of the first author was supported by the Hungarian-Austrian governmental scientific and technological cooperation.

Research of the second author was supported by the Hungarian National Foundation for Scientific Research Grant No. 16791/95.

Article copyright:
© Copyright 1999
American Mathematical Society