Types Of Indeterminate Equation Integer Solution

Diophantine equation is an important subject in number theory.It was greatly connected with algebra, combinatorial mathematics and computer science etc.The achievements in diophantine equations play an important role both in every branch of mathematics and in other subjects ,such as physics,economics.So there are still many people who have great interested in diophantine equations.The main work of this paper is to disscuss these Diophantine equations with the congruent method and the Algebraic Number Theory.First, the integer solutions satisfying some conditions to the equation x~2+ D=4y~3(D=11,-5,-13,-21,-29) were given. In this part,we will prove that the equation x~2+11=4y~3 has no integer solutions ;the equation x~2-5=4y~3 has integer solutions (x,y)-(±1,-1), (±3,1);the equation x~2-13=4y~3 has integer solutions (x,y)=(±3,-1); the equation x~2-21= 4y~3 has integer solutions (x,y)=(##5, l);the equation x~2-29=4y~3 has integer solutions (x, y)=(±0,1).Second, the integer solutions satisfying some conditions to the Diophantine equation x~2+C= y~n(C=2,n=3;C=1,n=5;C=64, nE=3) were given. In this part,we will prove that the equation x~2+2=y~3 has integer solutions (x, y)=(±5, 3);the equationx~2+1=y~5 has integer solutions (x, y)=(0,1);the equation x~2+64=y~3 has no integer solutions.