Pell方程组(a^(2)+4)x^(2)-y^(2)=4和x^(2)-bz^(2)=1的解  

Complete solutions of the simultaneous Pell’s equations(a^(2)+4)x^(2)-y^(2)=4 and x^(2)-bz^(2)=1

作　　者：罗钧文 程开敏[1] LUO Junwen;CHENG Kaimin(School of Mathematics and Information,China West Normal University,Nanchong,Sichuan 637092,China)

机构地区：[1]西华师范大学数学与信息学院,四川南充637092

出　　处：《内江师范学院学报》2025年第2期36-42,共7页Journal of Neijiang Normal University

基　　金：国家自然科学基金项目(11871058);四川省教育厅重点科研项目(16ZA0173);西华师范大学国家基金培育项目(22KA018)。

摘　　要：利用因式分解、二次剩余、Lehmer序列和Lehmer伴随序列的基本性质等初等方法,得到了Pell方程组(a^(2)+4)x^(2)-y^(2)=4和x^(2)-bz^(2)=1有正整数解的充分必要条件。进而证明了该方程组至多有一组正整数解,且当解存在时,求出了该方程组的全部解。证明了除三种特殊解外,这个唯一解由该方程组的第一个方程最小解的三次方或五次方给出。That the necessary and sufficient conditions that the simultaneous Pell equations(a^(2)+4)x^(2)-y^(2)=4 and x^(2)-bz^(2)=1 have positive integer solutions are obtained by using only the elementary methods of factorization,congruence,the quadratic residue and fundamental properties of Lehmer sequence and the associated Lehmer sequence.Moreover,that these simultaneous Pell equations have at most one solution in positive integers are proved.When a solution exists,the only solution of the system is given by the cube or 5 th power of the minimal positive integer solution of the first equation of the title equations except for the three special solutions.

关 键 词：丢番图方程 联立Pell方程 最小解 Lehmer序列 

分 类 号：O156.3[理学—数学]