关于Pell方程px^2-(pn±2)y^2=±1(p≡-1,±3(mod8)是素数)

On Pell Equation px^2-(pn±2)y^2=±1(p≡-1,±3(mod8),p is a prime factor)

Pell方程ax2-by2=±1(a,b∈Z+,ab不是完全平方数)可解性的判别是一个非常有意义的问题.运用Legendre符号和同余的性质给出了形如px2-(pn±2)y2=±1(p≡-1,±3(mod8)是素数)型Pell方程无正整数解的6个结论.这些结论对研究狭义Pell方程x2-Dy2=±1(D是非平方的正整数)起了重要作用.

The solubility of Pell equation ax2-by2=±1（a,b∈Z＋,ab is a non-square positive integer）is a very meaningful question.In this paper,by applying related knowledge of Legendre sign and nature of congruence,it works out six conclusions to judge that the sets of Pell equations such as px2-（pn±2）y2=±1（p≡-1,±3（mod 8）,and p is a prime number） have not positive integer solutions.These conclusions play an important role in the research on restricted Pell equation x2-Dy2=±1（D is a non-square positive integer）.