The Research Of The Two Kinds Of The Integer Solution Of Indeterminate Equation

Number theory is the mathematical discipline of a shining pearl, it has a longerhistory than Chinese history. Indefinite Equation is one of the important contents inthe number theory. The main structure is divided into two parts:The first partdiscussion of the integer solution of D=1, n=3, C=1; n=3,C=7；n=4,C=1in theequationD Dx~2+C=my~n.The following results are proved Equation x~2+5=y~3;x~2-7=y~3and equation x~2+1=19y~4has no integer solution. The second partdiscussion of the integer solution of n=3,D=7；n=3,D=-37；n=3,D=-85in theequationx2D4yn, the following results are proved:equation x~2+D=4y~nonly has integer solution (5,2)； Equation x~2-37=4y~3and equationx~2-85=4y~3has no integer solution. In this paper, on the basis of previous I amsolution these indefinite equations with congruence method and algebraic numbertheory method. The equation x~2+D=4y~n although did not get the most satisfactoryresults, but the D values more advanced one step, also Its provides a a direction that Iwill continue to engage in the study of number theory.