On the Diophantine equation x~p-1=Dy~n;

On the solution of the Diophantine equations x~2-2p=y~n;

On the Diophantine equation(15n)~x+(112n)~y=(113n)~z;

On the Diophantine equations x~4±y~6=z~2 and x~2+y~4=z~6;

When p is a odd prime and p ≠1 (mod 8), we get all solutions of diophantine equations ( x(x+1)(2x+1)=2p~ky~(2n) ) with elementary theory of number.

With the help of the elementary theory of number and Fermat method of infinite descent,some necessary conditions have been proved provided that the Diophantine equations x 4+mx 2y 2+ny 4=z 2 has positive Integer solutions that fit (x,y) =1 m.

Let p>3 be a prime integer prime,when the elementary grade method and the Diophantus Equation theories are used.

In this paper two theorems are given by using matrixvector description of polynomial multiplication, which are useful to resolve the Diophantus equation.

For combinatorics, linear Diophantine equations,counting integer points in polytope,Frobenins problemes etc.

With the rapid development of computer science in many fields, the peoblem of linear Diophantine equations has attracted more attention again.