On the Diophantine equations x2 ? 47y2 = 1 and y2 ? Pz2 = 49

2018 International Conference on Advanced Materials, Intelligent Manufacturing and Automation

材料科学;机械制造;运输工程

Zhao, Jianhong^1 ; Yang, Lixing^1

Department of Teachers and Education, Lijiang Teachers College, Lijiang, YUNNAN

674199, China^1

关键词: Diophantine equation;    Integer solutions;    Odd prime;   

Others  :  https://iopscience.iop.org/article/10.1088/1757-899X/382/5/052037/pdf
DOI  :  10.1088/1757-899X/382/5/052037

学科分类：材料科学（综合）

来源: IOP

【 摘 要 】

The integer solution of diophantine equations x2 - D1y2 = m, (D1 ∈ Z+ , m ∈ Z) and y2 - D2z2 = n, (D2 ∈ Z+ , n ∈ Z) is a matter of great concern. Researchers study for different m, n and D1 , D2 , and obtained some correlation results as follows. When m = 1, n = 1, the diophantine equations turns into x2 - D1y2 = 1 and y2 - D2z2 = 1. At present, there are only a few conclusions on it, see Ref [1] and [2]. When m = 1, n = 4, the integer solution see Ref [3] - [9]. When m = 1, n = 16, the previous conclusions see Ref [10]. When m = 1, n = 49, the diophantine equations turns into x2 - D1y2 = 1 and y2 - D2z2 = 49. In this case, D1 = 47, D2 can be expressed as 2 tp1a1 p2 a2 p3 a3 p4 a4 where ai = 0 or 1 for 1 ≤ i ≤ 4, and t ∈ Z+ , p s (1 ≤ s ≤ 4) are different odd primes. Up to now, there is no relevant result on the integer solution of x2 - 47y2 = 1 and y2 - Pz2 = 49, this paper mainly discusses the integer solution of it.