A parametric family of quartic Thue equations

doi:10.3868/s140-DDD-023-0016-x

Front. Math. China

2023, Vol. 18

Issue (3) : 147-163 https://doi.org/10.3868/s140-DDD-023-0016-x

RESEARCH　ARTICLE

A parametric family of quartic Thue equations

Zhigang LI¹(

), Pingzhi YUAN²

¹. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
². School of Mathematics, South China Normal University, Guangzhou 510631, China

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Abstract

In this paper,we give all primitive solutions of a parameterized family of quartic Thue equations:

　　　　　 $x 4 − 4 c x 3 y + (6 c + 2) x 2 y 2 + 4 c x y 3 + y 4 = 96 c + 169, c > 0.$

Keywords Extension of classical Legendre's theorem Baker-Wüstholz's theorem Thue equation

Corresponding Author(s): Zhigang LI

Online First Date: 16 November 2023 Issue Date: 07 December 2023

Cite this article:

Zhigang LI,Pingzhi YUAN. A parametric family of quartic Thue equations[J]. Front. Math. China, 2023, 18(3): 147-163.

URL:

https://academic.hep.com.cn/fmc/EN/10.3868/s140-DDD-023-0016-x
https://academic.hep.com.cn/fmc/EN/Y2023/V18/I3/147

1	A Baker. Contributions to the theory of Diophantine equations I, on the representation of integers by binary forms. Philos Thans Roy Soc London Ser A 1968; 263: 273–291
2	A Baker, H Davenport. The equation 3x2−2=y2 and 8x2−7=z2. Quart J Math Oxford 1969; 20(2): 129–137
3	A Baker, G Wüstholz. Logarithmic forms and group varieties. J Reine Angew Math 1993; 442: 19–62
4	M A Bennett. On the number of solutions of simultaneous Pell equations. J. Reine Angew Math 1998; 498: 173–199
5	Y Bilu, G Hanrot. Solving Thue equations of high degree. J Number Theory 1996; 60: 373–392
6	A Dujella, A Pethö. Generalization of a theorem of Baker and Davenport. Quart J Math Oxford 1998; 49: 291–306
7	A Dujella, B Jadrijević. A parametric family of quartic Thue equations. Acta Arith 2002; 101: 159–170
8	A Dujella. A family of quartic Thue inequalities. Acta Arith 2004; 111: 61–76
9	A Dujella. Continued fractions and RSA with small secret exponent. Tatra Mt Math Publ 2004; 29: 101–112
10	P Fatou. Sur l’approximation des incommenurables et les series trigonometriques. C R Acad Sci Paris 1904; 139: 1019–1021
11	I Gaál. Diophantine Equations and Power Integral Bases: New computational Methods. New York: Springer-Verlag, 2002
12	C Heuberger, A Pethö, R F Tichy. Complete solution of parametrized Thue equations. Acta Math Inform Univ Ostraviensis 1998; 6: 93–113
13	L K Hua. Introduction to Number Theory. New York: Springer, 1982
14	Z KeQ Sun. Talking about Indeterminate Equations. New York: Shanghai Educational Publishing House, 1980
15	A Pethö, R Schulenberg. Effectives Lösen von Thue Gleichungen. Publ Math Debrecen 1987; 34: 189–196
16	E Thomas. Complete solutions to a family of cubic Diophantine equations. J Number Theory 1990; 34: 235–250
17	A Thue. Über Annäherungswerte algebraischer Zahlen. J Reine Angew Math 1909; 135: 284–305
18	N Tzanakis, B M M de Weger. On the Practical Solution of the Thue Equation. J Number Theory 1989; 31: 99–132
19	N Tzanakis. Explicit solution of a class of quartic Thue equations. Acta Arith 1993; 64: 271–283
20	R T Worley. Estimating \|α−p/q\|. J Austral Math Soc 1981; 31: 202–206

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