Extremal families of cubic Thue equations
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, p. 389-403

We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form F(x,y)=1 with at least 5 such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction.

Nous déterminons les solutions entières d’une nouvelle famille infinie d’équations de Thue cubiques, chacune de ces équations ayant exactement cinq solutions. Notre approche combine des arguments élémentaires avec des limites inférieures pour les formes linéaires en logarithmes et la réduction L 3 .

DOI : https://doi.org/10.5802/jtnb.907
Classification:  11D25,  11E76
@article{JTNB_2015__27_2_389_0,
     author = {Bennett, Michael A. and Ghadermarzi, Amir},
     title = {Extremal families of cubic Thue equations},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     pages = {389-403},
     doi = {10.5802/jtnb.907},
     mrnumber = {3393160},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2015__27_2_389_0}
}
Bennett, Michael A.; Ghadermarzi, Amir. Extremal families of cubic Thue equations. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 389-403. doi : 10.5802/jtnb.907. http://www.numdam.org/item/JTNB_2015__27_2_389_0/

[1] S. Akhtari, Cubic Thue Equations. Publ. Math. Debrecen. 75 (2009), 459–483. | MR 2588218 | Zbl 1199.11076

[2] A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms. Philos. Trans. Roy. Soc. London Ser. A 263 (1967/1968), 173–191. | MR 228424 | Zbl 0157.09702

[3] A. Baker and H. Davenport, The equations 3x 2 -2=y 2 and 8x 2 -7=z 2 . Quart. J. Math. Oxford 20 (1969), 129–137. | MR 248079 | Zbl 0177.06802

[4] V.I. Baulin, On an indeterminate equation of the third degree with least positive discriminant (Russian). Tul’sk Gos. Ped. Inst. Ucen. Zap. Fiz. Math. Nauk. Vip. 7 (1960), 138–170. | MR 199149

[5] K. Belabas, A fast algorithm to compute cubic fields. Math. Comp. 66 (1997), 1213–1237. | MR 1415795 | Zbl 0882.11070

[6] K. Belabas and H. Cohen, Binary cubic forms and cubic number fields. Proceedings Organic Mathematics Workshop, Vancouver 1995 (CMS Conference Proceedings 20, 1997), 175–204. | MR 1483919 | Zbl 0904.11044

[7] K. Belabas and H. Cohen, Binary cubic forms and cubic number fields. Proceedings of a Conference in Honor of A.O.L. Atkin, 1995 (AMS/IP Studies in Advanced Mathematics 7, 1998), 191–219. | MR 1486838 | Zbl 0915.11024

[8] M.A. Bennett, On the representation of unity by binary cubic forms. Trans. Amer. Math. Soc. 353 (2001), 1507–1534. | MR 1806730 | Zbl 0972.11014

[9] E. Bombieri and W. Schmidt, On Thue’s equation. Invent. Math. 88 (1987), 69–82. | MR 877007 | Zbl 0614.10018

[10] A. Bremner, Integral generators in a certain quartic field and related Diophantine equations. Michigan Math. J. 32 (1985), 295–319. | MR 803834 | Zbl 0585.12005

[11] B.N. Delone, Über die Darstellung der Zahlen durch die binäre kubischen Formen von negativer Diskriminante. Math. Z. 31 (1930), 1–26. | MR 1545095

[12] L.E. Dickson, History of the Theory of Numbers. Chelsea, New York, 1971.

[13] J.H. Evertse, Upper Bounds for the Numbers of Solutions of Diophantine Equations. Thesis, Amsterdam, 1983. | MR 726562 | Zbl 0517.10016

[14] I. Gaál and N. Schulte, Computing all power integral bases of cubic fields. Math. Comp. 53 (1989), 689–696. | MR 979943 | Zbl 0677.10013

[15] C. Heuberger, On families of parametrized Thue equations. J. Number Theory 76 (1999), 45–61. | MR 1688196 | Zbl 0934.11012

[16] C. Heuberger, On general families of parametrized Thue equations. Algebraic Number Theory and Diophantine Analysis. Proceedings of the International Conference held in Graz, Austria, 1998 (F. Halter-Koch and R. F. Tichy, eds.), Walter de Gruyter, (2000), 215–238. | MR 1770464 | Zbl 0963.11019

[17] C. Heuberger, A. Togbé and V. Ziegler, Automatic solution of families of Thue equations and an example of degree 8. J. Symbolic Computation 38 (2004), 145–163. | MR 2093887 | Zbl 1137.11307

[18] E. Lee, Studies on Diophantine Equations. PhD Thesis, Cambridge University, 1992.

[19] F. Lippok, On the representation of 1 by binary cubic forms of positive discriminant. J. Symbolic Computation 15 (1993), 297–313. | MR 1229637 | Zbl 0780.11012

[20] W. Ljunggren, Einige Bemerkungen über die Darstellung ganzer Zahlen durch binäre kubische Formen mit positiver Diskriminante. Acta Math. 75 (1942), 1–21. | MR 17303

[21] K. Mahler, On Thue’s theorem. Math. Scand. 55 (1984), 188–200. | MR 787196 | Zbl 0544.10014

[22] E. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Math. 64 (2000), 1217–1269. | MR 1817252 | Zbl 1013.11043

[23] M. Mignotte, Pethő’s cubics. Publ. Math. Debrecen 56 (2000), 481–505. | MR 1765995 | Zbl 0960.11020

[24] M. Mignotte, Linear forms in two and three logarithms and interpolation determinants. (English summary). Diophantine equations, 151–166, Tata Inst. Fund. Res. Stud. Math. 20, Tata Inst. Fund. Res., Mumbai, (2008). | MR 1500224 | Zbl 1198.11071

[25] M. Mignotte and N. Tzanakis, On a family of cubics. J. Number Theory 39 (1991), 41–49. | MR 1123167 | Zbl 0734.11025

[26] T. Nagell, Darstellung ganzer Zahlen durch binäre kubische Formen mit negativer Diskriminante. Math. Zeitschr. 28 (1928), 10–29. | MR 1544935

[27] T. Nagell, Remarques sur une classe d’équations indéterminées. Arkiv för Math. 8 (1969), 199–214. | MR 271019 | Zbl 0214.29901

[28] R. Okazaki, Geometry of a cubic Thue Equation. Publ. Math. Debrecen. 61 (2002), 267–314. | MR 1943695 | Zbl 1012.11022

[29] A. Pethő, On the representation of 1 by binary cubic forms with positive discriminant. Number Theory, Ulm 1987 (Springer LMN 1380), 185–196. | MR 1009801 | Zbl 0677.10012

[30] A. Pethő and R. Schulenberg, Effektives Lösen von Thue Gleichungen. Publ. Math. Debrecen (1987), 189–196. | MR 934900 | Zbl 0657.10015

[31] E. Thomas, Complete solutions to a family of cubic Diophantine equations. J. Number Theory 34 (1990), 235–250. | MR 1042497 | Zbl 0697.10011

[32] E. Thomas, Solutions to certain families of Thue equations. J. Number Theory 43 (1993), 319–369. | MR 1212687 | Zbl 0774.11013

[33] A. Thue, Über Annäherungenswerte algebraischen Zahlen. J. reine angew. Math. 135 (1909), 284–305. | MR 1580770

[34] N. Tzanakis, The diophantine equation x 3 -3xy 2 -y 3 =1 and related equations. J. Number Theory 18 (1984), 192–205. | MR 741950 | Zbl 0552.10008

[35] N. Tzanakis and B.M.M. de Weger, On the practical solution of the Thue equation. J. Number Theory 31 (1989), 99–132. | MR 987566 | Zbl 0657.10014

[36] I. Wakabayashi, On a family of cubic Thue equations with 5 solutions. Acta Arith. 109 (2003), 285–298. | MR 1980263 | Zbl 1031.11018

[37] V. Ziegler, Thomas’ conjecture over function fields. J. Th. Nombres Bordeaux 19 (2007), 289–309. | MR 2332067 | Zbl 1193.11030