摘要
设a是正整数.本文证明了:当a=1时,方程X^2-(a^2+1)Y^4=8~6a仅有正整数解(X,Y)=(2,1);当a=2时,该方程仅有正整数解(X,Y)=(1,1);当a=3时,该方程无正整数解(X,Y);当a=4时,该方程仅有2组互素的正整数解(X,Y)=(1,1)和(103,5);当a≥5且6a+1非平方数时,该方程最多有3组互素的正整数解(X,Y);当a≥5且6a+1为平方数时,该方程最多有4组互素的正整数解(X,Y).
Let a be a positive integer.In this paper,we prove that if a = 1,then the equation X^2-（a^2 ＋ 1）Y^4 = 8-6a has only one positive integer solution（X,Y） =（2,1）;if a = 2,then the equation has only one positive integer solution（X,Y） =（1,1）;if a = 3,then the equation has no positive integer solution（X,Y）;if a = 4,then the equation has only two coprime positive integer solutions（X,Y） =（1,1）,（103,5）;if a ≥ 5 and 6a ＋ 1 is a nonsquare positive integer,then the equation has at most three coprime positive integer solutions（X,Y）;if a ≥ 5 and 6a ＋ 1 is a square,then the equation has at most four coprime positive integer solutions（X,Y）.
出处
《数学进展》
CSCD
北大核心
2017年第3期355-372,共18页
Advances in Mathematics（China）
基金
国家自然科学基金(No.11471144)
云南省教育厅科研课题(No.2014Y462)
泰州学院重点课题(No.TZXY2014ZDKT007)
泰州学院教博课题(No.TZXY2016JBJJ001)
关键词
四次方程
虚二次域
丢番图逼近
解数
上界
quartic equations
imaginary quadratic fields
Diophantine approximations
number of positive integer solutions
upper bound
