摘要
设a是正整数,证明了当a=1时,方程X2-(a2+1)Y4=35-12a仅有正整数解(X,Y)=(5,1);当a=2时,该方程仅有正整数解(X,Y)=(4,1)和(56,5);当a=3时,该方程仅有正整数解(X,Y)=(3,1);当a=4时,该方程仅有正整数解(X,Y)=(2,1)和(202,7);当a=5时,该方程仅有1组互素的正整数解(X,Y)=(1,1);当a=6时,该方程无正整数解(X,Y);当a≥7且12a+1为非平方数时,该方程最多有3组互素的正整数解(X,Y);当a≥7且12a+1为平方数时,该方程最多有4组互素的正整数解(X,Y).
Let abe an positive integer.We prove that if a=1,then the equation X^2-（a^2＋1）Y^4=35-12 ahas only one positive integer solution（X,Y）=（5,1）;If a=2,then the equation has only two positive integer solutions,（X,Y）=（4,1）and（56,5）;If a=3,then the equation has only one positive integer solution（X,Y）=（3,1）;If a=4,then the equation has two positive integer solutions（X,Y）=（2,1）and（202,7）;If a=5,then the equation has one coprime positive integer solution（X,Y）=（1,1）;If a=6,then the equation has no positive integer solution（X,Y）;If a≥7and 12a＋1is a nonsquare positive integer,the equation has at most three coprime positive integer solutions;While if a≥7and 12a＋1is a square,the equation has at most four coprime positive integer solutions.
出处
《浙江大学学报（理学版）》
CAS
CSCD
北大核心
2016年第2期138-143,共6页
Journal of Zhejiang University（Science Edition）
基金
江苏省教育科学"十二五"规划项目(D201301083)
云南省教育厅科研项目(2014Y462)
泰州学院教授基金项目(TZXY2015JBJJ002)
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
