Study Of Several Special Kinds Of Diophantine Equations

Another name of Diophantine equation is indeterminate equation. It's an important branch of the number theory, and also the active one in the field of mathematics through the whole history. In the recent ten years, Diophantine equation not only develops itself very actively, but also promotes the development of other fields. Therefore, there are so many working mathematicians who are interested in studying them.This paper is to study several special kinds of Diophantine equation which contains the following five aspects:1. The author first introduced the research progress of Diophantine Equation x~3±1 = Dy~2. Using congruence and recurrent sequence, the author proved that the Diophantine Equation x~3 + 1 = 86y~2 has the only integer solution (x, y) = (-1,0),(7,±2).2. Next the author introduced the research progress of Diophantine Equation Ax~2 + B = y~n. Using the method of algebraic number theory, the author has proved that the diophantine equation x~2 + 16 = y~7 has no integer solution and so does thatDiophantine equation x~2 + 25 = y~3.3. Diophantine Equation(?)=(?)has been demonstrated by usinginequality method. 4. The author was inspired from Diophantine equations (?), struc-tured two interesting Diophantine equations, and then provided the solutions.5. Finally the author studied the equation which contains a number-theoretic function.It will help the study of other Diophantine equations, provide some references and make some contributions to the research of Diophantine equation by the solutions of all these problems.