On the system of Diophantine equations (m2-1)r + b2 = c2 and (m2-1)x + by = cz

JOURNAL OF NUMBER THEORY

卷:153

Article

Miyazaki, Takafumi¹  Luca, Florian² 

[1] Nihon Univ, Dept Math, Coll Sci & Technol, Chiyoda Ku, Tokyo 1018308, Japan

[2] Univ Witwatersrand, John Knopfmacher Ctr Applicable Anal & Number The, Sch Math, ZA-2050 Johannesburg, South Africa

关键词: Exponential Diophantine equation;    Linear form in logarithms of algebraic numbers;   

DOI  :  10.1016/j.jnt.2014.12.021

来源: Elsevier

【 摘 要 】

Given positive integers r and m, one can create a positive integer solution (b, c) to the first equation in the title by setting b and c as 2b = (m + 1)(r) - (m - 1)(r) and 2c = (m + 1)(r) + (m - 1)(r). In this note we show that there are only finitely many pairs (r, m) with r = 2 (mod 4) and m even such that the second equation in the title holds for some triple (x, y, z) of positive integers with (x, y, z) not equal (r, 2, 2). (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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