摘要
利用初等的证明方法即同余法、Pell方程的整数解的性质、Maple小程序以及递归序列和二次剩余的方法,对一个丢番图方程x3+1=57y2的整数解进行了研究。证明过程中仅涉及到初等的数论知识,首先利用等式的性质把原丢番方程的解转化为4种情形进行讨论;对其第一种利用等式的性质得出无整数解,第二种情形利用同余式得出无整数解,后面两种利用同余式递归数列和平方剩余的相关知识以及maple小程序得出整数解和平凡解;最后综合得该丢番图方程仅有整数解(x,y)=(-1,0),(8,±3)。
In this paper the author studies all the integer solutions to the diophantine equation x^3 ＋ 1 = 57y^2. The process is as follows ： classify the will-be integer solutions to the diophantine equation into four equations by equation firstly, then take models on these equations, the first equation and second equantion aren＇t integer solutions. At last two equations are integer solutions, at the same time, the methods of recursive sequences and maple formality and Pell equation and quadratic remainder are used. At last, it is proved that the diophantine equation x^3 ＋ 1 = 57y^2 has only positive integer solutions （x,y） = （ - 1,0）, （ 8, ±3 ）.
出处
《重庆师范大学学报（自然科学版）》
CAS
2010年第3期41-43,72,共4页
Journal of Chongqing Normal University:Natural Science
基金
重庆邮电大学自然科学基金(No.A20080-40)
关键词
丢番图方程
整数解
递归序列
平方剩余
Diophantine equation
integer solution
recurrent sequence
guadratic remaider
