E. Thomas was one of the first to solve an infinite family of Thue equations, when he considered the forms and the family of equations , . This family is associated to the family of the simplest cubic fields of D. Shanks, being a root of . We introduce in this family a second parameter by replacing the roots of the minimal polynomial of by the -th powers of the roots and we effectively solve the family of Thue equations that we obtain and which depends now on the two parameters and .
E. Thomas fut l’un des premiers à résoudre une famille infinie d’équations de Thue, lorsqu’il a considéré les formes et la famille d’équations , . Cette famille est associée à la famille des corps cubiques les plus simples de D. Shanks, étant une racine de . Nous introduisons dans cette famille un second paramètre en remplaçant les racines du polynôme minimal de par les puissances -ièmes des racines et nous résolvons de façon effective la famille d’équations de Thue que nous obtenons et qui dépend maintenant des deux paramètres et .
Keywords: Simplest cubic fields, family of Thue equations, diophantine equations, linear forms of logarithms.
@article{JTNB_2015__27_2_537_0, author = {Claude Levesque and Michel Waldschmidt}, title = {A family of {Thue} equations involving powers of units of the simplest cubic fields}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {537--563}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {27}, number = {2}, year = {2015}, doi = {10.5802/jtnb.913}, mrnumber = {3393166}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/} }
TY - JOUR AU - Claude Levesque AU - Michel Waldschmidt TI - A family of Thue equations involving powers of units of the simplest cubic fields JO - Journal de théorie des nombres de Bordeaux PY - 2015 SP - 537 EP - 563 VL - 27 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/ DO - 10.5802/jtnb.913 LA - en ID - JTNB_2015__27_2_537_0 ER -
%0 Journal Article %A Claude Levesque %A Michel Waldschmidt %T A family of Thue equations involving powers of units of the simplest cubic fields %J Journal de théorie des nombres de Bordeaux %D 2015 %P 537-563 %V 27 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/ %R 10.5802/jtnb.913 %G en %F JTNB_2015__27_2_537_0
Claude Levesque; Michel Waldschmidt. A family of Thue equations involving powers of units of the simplest cubic fields. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 537-563. doi : 10.5802/jtnb.913. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.913/
[1] J.H. Chen, Efficient rational approximation of a class of algebraic numbers, (in Chinese) Chinese Science Bulletin, 41 (1996), 1643–1646. | MR
[2] G. Lettl, A. Pethő, and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc., 351 (1999), 1871–1894. | MR | Zbl
[3] C. Levesque and M. Waldschmidt, Families of cubic Thue equations with effective bounds for the solutions Springer Proceedings in Mathematics & Statistics 43 (2013) 229—243. | MR
[4] C. Levesque and M. Waldschmidt, Solving simultaneously Thue Diophantine equations: almost totally imaginary case, Proceedings of the International Meeting on Number Theory 2011, in honor of R. Balasubramanian, Lecture Notes Series in Ramanujan Mathematical Society, to appear. | MR
[5] C. Levesque and M. Waldschmidt, Solving effectively some families of Thue Diophantine equations, Moscow J. of Combinatorics and Number Theory, 3 3–4 (2013), 118–144. | MR
[6] C. Levesque and M. Waldschmidt, Families of Thue equations associated with a totally real rank subgroup of the units of a number field. Work in progress.
[7] M. Mignotte, Verification of a conjecture of E. Thomas, J. Number Theory, 44 (1993), 172–177. | MR | Zbl
[8] M. Mignotte, A. Pethő and F. Lemmermeyer, On the family of Thue equations , Acta Arith. 76 (1996), 245–269. | MR | Zbl
[9] T.N. Shorey and R. Tijdeman, Exponential Diophantine equations, vol. 87 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1986. | MR | Zbl
[10] E. Thomas, Complete solutions to a family of cubic Diophantine equations, J. Number Theory, 34 (1990), pp. 235–250. | MR | Zbl
[11] I. Wakabayashi, Simple families of Thue inequalities, Ann. Sci. Math. Québec 31 (2007), no. 2, 211–232. | MR | Zbl
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