不定方程x^2+4^n=y^13整数解的完全刻画  

Complete Characterization for the Integer Solution of the Diophantine Equation x^2+4^n=y^13

作　　者：钟焕旻 黄宇飞[3] ZHONG Huan-min;HUANG Yu-fei(School of Mathematical Sciences,South China Normal University,Guangzhou,Guangdong,510631,China;Xin’an Middle School(Group)Second Foreign Language School,Shenzhen,Guangdong,518100,China;Department of Mathematics Teaching,Guangzhou Civil Aviation College,Guangzhou,Guangdong,510403,China)

机构地区：[1]华南师范大学数学科学学院,广东广州510631 [2]新安中学(集团)第二外国语学校,广东深圳518100 [3]广州民航职业技术学院数学教学部,广东广州510403

出　　处：《新疆师范大学学报（自然科学版）》2020年第2期34-38,共5页Journal of Xinjiang Normal University(Natural Sciences Edition)

基　　金：国家自然科学基金项目(11501139);广东省普通高校特色创新项目(2019GKTSCX001);广航院国基校级重点培育项目(18X0429)资助。

摘　　要：文章研究指数型Lebesgue-Nagell不定方程x^2+B=y^k的整数解是数论中的一类重要课题,其中B是非负整数,k是正整数。应用代数数论的方法完全刻画了不定方程x^2+4^n=y13的整数解,既证明了不定方程x^2+4^n=y^13有整数解(当且仅当n≡0,6(mod 13)),且其整数解分别为(n,x,y)=(13m,0,4^m)或(13m+6,±2^{13m+6},2^{2m+1}),其中n,m是非负整数.The characterization for the integer solution of the exponential Lebesgue-Nagell dio-phantine equation x^2+B=yk is an important subject in number theory,where B is a nonnegative integer and k is a positive integer.In this paper,we give a complete characterization for the integer solution of the diophantine equation x^2+4^n=y^13 by using algebraic number theory,i.e.,the diophantine equation x^2+4^n=y^13 has only integer solutions(n,x,y)=(13m,0,4^m)or(13m+6,±2^13m+6,2^2m+1),where n,m are nonnegative integers.

关 键 词：不定方程 整数解 代数数论 

分 类 号：O156.2[理学—数学]