On a class of quartic Thue equations with three parameters

JOURNAL OF NUMBER THEORY

卷:202

Article

He, Bo¹  Odjoumani, Japhet²  Togbe, Alain³ 

[1] China West Normal Univ, Sch Math & Informat, Nanchong 637000, Sichuan, Peoples R China

[2] Univ Abomey Calavi, Inst Math & Sci Phys, Dangbo, Benin

[3] Purdue Univ Northwest, Dept Math Stat & Comp Sci, 1401 S,US 421, Westville, IN 46391 USA

关键词: Thue equations;    Method of Tzanakis;    System of pellian equations;    Linear forms in logarithms;   

DOI  :  10.1016/j.jnt.2019.01.002

来源: Elsevier

【 摘 要 】

Let k, m and n be integers. In this paper, for a fixed integer mu not equal 0, we show that the family of Thue equation x(4) - kmnn(3)y + (km(2) - km(2) + 2)x(2)y(2) + kmnxy(3) + y(4) = mu is reducible by Tzanakis's method into a system of pellian equations kV(2) - (km(2) + 4)U-2 = -4 mu, kZ(2) - (kn(2) - 4)U-2 = 4 mu, with any triple of integers (k, m, n) such that k > 0, vertical bar n vertical bar >= 2, vertical bar m vertical bar >= 2. We consider this system for any even integer k not equal 2 square, mu = 1 and we prove that for all integers vertical bar n vertical bar >= 2 and vertical bar m vertical bar >= 2 that are sufficiently large and have sufficiently large common divisor this system has only the trivial solutions (U, V, Z,) = ( 1, m, n). We also show that if k not equal 2 square is even, then the system has in general at most 8 solutions in positive integers. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free