Extremal families of cubic Thue equations

Bennett, Michael A.; Ghadermarzi, Amir

Journal de théorie des nombres de Bordeaux, Tome 27 (2015) no. 2, pp. 389-403.

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 $L^3$.

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.

Classification : 11D25, 11E76

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

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

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

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