The Solvable Research On Two Kinds Of Classical Diophantine Equations

The Diophantine equation means that the number of unknown numbers is more than the number of equations,and the value of unknown number is limited to positive integer,integer,rational numbers and so on,whose is coefficient polynomial equation,it is also known as the indefinite equation.There are three main problems to be solved in the study of the Diophantine equation:one is to determine when there is a solution,the other is to determine the number of solutions when there are solutions,and the third is to find out all of the solutions.Because the Diophantine equation is very rich,but there is no unified way to obtain solution,it determines the difficulty of its research,but at the same time attracts many scholars to make use of various possible methods to study its solvability.In this paper,by using the method of elementary number theory and on the basis of previous studies,the solvability problem of the exponent Diophantine equation and the cubic Diophantine equation are discussed,all the solutions of these equations are given and the results of previous studies are generalized.The specific contents are as follows:(1)We study the solvability of the exponential Diophantine equation X2+28=y11,and generalize the solvability of the more general exponential Diophantine equation X2+28=y2m+5.It is proved that the equation has no positive integer solution.(2)We study the solvability of the exponential Diophantine equation(2k)x+by=(b+2k)z.It is proved that the equation has only a unique positive integer solution.(3)We study the solvability of the cubic Diophantine equation x3+93=18y2.It is proved that the equation has only two positive integer solutions.(4)We study the solvability of the cubic Diophantine equation x3?1=2p1p2Qy2.It is proved that the equation has no positive integer solution.Finally,we summarize the problems about the solving of the indeterminate equations,and put forward the direction that we should make further efforts to study it.