On the system of Diophantine equations (m^2-1)^r+b^2=c^2 and (m^2-1)^x+b^y=c^z
On the system of Diophantine equations (m^2-1)^r+b^2=c^2 and (m^2-1)^x+b^y=c^z
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关于丢番图方程组 (m^2-1)^r b^2=c^2 和 (m^2-1)^x b^y=c^z
DOI：
10.1016/j.jnt.2014.12.021
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发表时间：
2015
影响因子：
0.7
通讯作者：
Florian Luca
中科院分区：
文献类型：
--
作者：
Takafumi Miyazaki;Florian Luca
给定正整数r和m，可以通过将B和c设置为2 B=（m+ 1）r−（m− 1）r和2 c=（m +1）r+（m− 1）r来创建标题中第一个方程的正整数解（B，c）。在本文中，我们证明了只有1000对（r，m）满足r <$2（mod 4）和m偶数，使得标题中的第二个方程对某个正整数三元组（x，y，z）满足（x，y，z）<$（r，2，2）成立.
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 2 b=(m+ 1) r−(m− 1) r and 2 c=(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)≠(r, 2, 2).