Research On Several Classes Of Diophantine Equation

In the preface,this paper briefly introduces the background and research status of Diophantine equations.The second chapter gives some basic concepts,properties and theorems needed in this paper.On the basis of predecessors,this paper mainly solves some cubic Diophantine equations and high order Diophantine equations.For cubic Diophantine equationAx²+B=Cy³?A,B,C?Z?,this paper proves the Diophantine equationx²-45=y³has no integer solution,and the Diophantine equation111x²-27=y³has only integer solutions?x,y?=?-,10?,?11,±?2,the Diophantine equation485x²-27=y^{3 has only the integer solution}?x,y?=?0,-3?,the Diophantine equation x²+B=4y³?B???1,2?mod 4??has no integer solutions.For high order Diophantine equationAx²+B=Cy⁷?A,B,C?Z?,this paper proves the Diophantine equationx²+256=y⁷and x²+1024=y⁷ have no integer solutions.