关于丢番图方程 kx^2-ly^2=4

On Diophantine Equation kx^2-ly^2=4

利用Stormer定理及其推广和Lehmer序列及其伴随序列的性质,讨论不定方程kx^2-ly^2=4的正整数解(x,y)与其最小解ε的关系。设xn√k+yn√l/2=(ε/2)^ n,我们证明了当x有且只有一个素因子不整除k时,除(k,l,x)=(5,1,x 25)外,x=x 1或x=x q;当x有且只有两个素因子不整除k时,除(k,l,x)=(5,1,x 125)或(k,l,x)=(5,1,x 5q)外,x=x 1或x=x q或x=x q^2。

In this paper,the relation between the positive integer solution(x,y)of Diophantine equations kx 2-ly 2=4 and its minimal solutionεis discussed by using Stormer theorem and its generalizations on Pell equation,and fundamental properties of Lehmer sequences and its associated sequences.Let xn√k+yn√l/2=(ε/2)^ n,it is proved that if there is only one prime divisor of x not dividing k,then x=x 1 or x=x q except for(k,l,x)=(5,1,x 25)and if there are only two prime divisors of x not dividing k,then x=x 1 or x=x q or x=x q 2 except for(k,l,x)=(5,1,x 125)or(k,l,x)=(5,1,x 5q).