关于两个Diophantine方程的求解

Solution of Two Diophantine Equations

对于部分无平方因子整数D,其二次域Q(D^(1/2))是Euclid域,那么它所对应的Euclid整环中算术基本定理成立。利用二次Euclid域的整除理论讨论了不定方程x^2±3=4y5,x,y∈Z的整数解情况,并得到了其所有整数解,即证明了不定方程x^2+3=4y^5,x,y∈Z仅有整数解(x,y)=(±1,1),而不定方程x^2-3=4y^5,x,y∈Z无整数解。

For some square-free integer D,its secondary domain Q（D^（1/2））is Euclid domain,then its corresponding Euclid fundamental theorem of arithmetic of integral domain is workable.This paper uses exact division theory in secondary Euclid domain to discuss the integral solution in indefinite equation x^2±3=4y^5,x,y ∈Z,obtaining its all integral solutions,which demonstrates that there is only one integral solution in indefinite equation x^2＋3=4y^5,x,y∈Z,while there is no integral solution in x^2-3=4y^5,x,y∈Z.