On The Integer Solutions Of Several Kinds Of Indefinite Equations

Diophantine equation, as a very important branch of number theory, has a long history and rich content. The Diophantine equation refers to the range of solutions of positive integer, integer, rational numbers or algebraic equation or integral equations, etc. Generally, the numbers of equation are much than unknown numbers of the equations. It is associated with algebraic number theory, the combination of design and finite simple groups.The main job of this thesis is to use algebraic and number theory, to discuss the integer solutions of the Diophantine equation.First of all, we will be discussing the integer solutions of the Diophantine equation x~2+D= y~3, (D=-14,-42). In this part, we will demonstrate that the equations x~2-14= y~3, x~2-42= y~3 have no integer solutions.Secondly, we will be discussing the integer solutions of the Diophantine equation x~2+D= 4y~3(D= l9, A3,61,15,35,51,79,71,-69). And we will demonstrate that the integer solutions of the equation x~2+D= 4y~3(D= 19,43,67,15,35,51,-69) have no integer solutions, the integer solutions of the equation x~2+79= Ay~3 is (x, y)= (±265,26),and the integer solutions of the equation x~2+71= 4y~3 is (x,y)= (±235,24).Finally, we will be discussing the integer solutions of the Diophantine equation x~2+D= 4y~5(D= 19,35,51,91,31,59,83). And we will demonstrate that the integer solutions of the equation x~2+D= 4y~5(D=19,35,51,91,31,59,83) have no integer solutions.