Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: January 26, 1999
MathSciNet review: 1487624
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [1] A. Baker, Rational approximations to $\sqrt[3 ]{2}$ 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 $X^{2} +1 = 2Y^{4}$, J. Number Theory 48 (1994), 62-74. MR 95i:11019
  • [4] Chen Jian Hua and P. M. Voutier, Complete solution of the Diophantine equation $X^{2} +1 = d Y^{4}$ 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 $6$ de $\mathbb Q$, 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 $\theta (x;3,l)$ and $\psi (x;3,l)$, 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 $x^{3}- \break (n-1)x^{2}y-(n+2)$ $xy^{2}-y^{3}=k$, 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 $\sqrt[3 ]{2}$ and other algebraic numbers revisited, Indag. Math. (to appear).

Similar Articles

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

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

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

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