摘要
设a≥2是正整数.本文证明了:当a=2时,方程X^2一(a^2+1)Y^4=3-4a仅有正整数解(X,Y)=(20,3);当a=3时,该方程仅有2组互素的正整数解(X,Y)=(1,1)和(79,5);当a≥4且4a+1非平方数时,该方程最多有4组互素的正整数解(X,Y);当a≥4且4a+1为平方数时,该方程最多有5组互素的正整数解(X,Y).
Let a ≥ 2 be a positive integer.In this paper,we will prove that if a = 2,then the equation X^2-(a^2 + 1)Y^4 = 3- 4a has only one positive integer solution(X,Y) =(20,3);if a = 3,then the equation has only two coprime positive integer solutions(X,Y) =(1,1),(79,5);if a 4 and 4a + 1 is a nonsquare positive integer,then the equation has at most four coprime positive integer solutions(X,Y);if a ≥ 4and 4a + 1 is a square,then the equation has at most five coprime positive integer solutions(X,Y).
出处
《数学学报(中文版)》
CSCD
北大核心
2016年第1期21-36,共16页
Acta Mathematica Sinica:Chinese Series
基金
江苏省教育科学十二五规划课题(D201301083)
云南省教育厅科研课题(2014Y462)
泰州学院重点课题(TZXY2014ZDKT007)
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