关于两个Diophantine方程的求解

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。

Solution of Two Diophantine Equations

For some square-free integer D ,its secondary domain Q ( D )is Euclid domain,then its corre-sponding 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 .