About the diophantine equation x~2-3y~4=97;

On Diophantine equation x~3+64=21y~2;

On the diophantine equation x~2+7=4y~3;

Successive solutions to dualistic simple indeterminate equation;

An integer solution to one kind to reciprocal indeterminate equation;

Fibonacci series and a positive integer solution to an indeterminate equation;

The Application of Generating Function in Solving Indefinite Equation;

The positive primitive solution of indefinite equation x~2+my~2=z~2,when m is a square free number;

In this paper,the author has proved that the diophantine equation x~3-27=7y~2 has only an integer solution(x,y)=(3,0).

In this paper,the author has proved that the Diophantine equation x~3-1=157y~2has only integer solution(x,y)=(1,0).

In this paper,the author has proved that the diophantine equationx 2+16 = y 7 has no integer solution.

Then with the help of these results, all solutions of several Diophantine equations in the rings of integers of two quadratic imaginary fields are determined, which implies that the Fermat equation has no non-trivial solutions in these rings.

By using model-theoretic methods,it is verified that one type of the diophantine equations a1xr11+a2xr22+…+anxrnn=bys(a1,a2,…,an,b are integers,r1,r2,…,rn,sare positive integers) have solutions.

All solutions of several important Diophantine equations are determined in a real quadratic field,which implies that the Fermat equation has no non-trivial solutions in this ring when n=4.

On the system of diophantine equations x~2-2y~2=1 and y~2-150z~2=4;

On the system of Diophantine equations (m+2)x 2-my 2=2, (4m+4)y 2-(m+2)z 2=3m+2;

In this paper,it has proved that the system of diophantine equations y~2-10x~2=9 and z~2-17x~2=16 has only integer solution x=0 by using the elementary methods.