1. Diophantine equation; prime; integer solution; Legendre symbol; congruence.

2. Diophantine equation; Positive integer solution; congruence; Recurrent sequence.

3. Diophantine equation is an important subject in number theory and closely connected with algebraic number theory, combinatorics, algebraic geometry and computer science etc.

4. 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.

5. An elementary proof of the Diophantine equation (The equation abbreviated) is given by using recursion sequence method.

6. In the recent ten years, Diophantine equation not only develops itself very actively, but also promotes the development of other fields.

7. On the diophantine equation is the number one branch, it has a long history and rich content.

8. So many researchers study the Diophantine Equation extensively and highly in the domestic and abroad.

9. The method does not need solve the Diophantine equation, but reduces the computation of the algorithm.

10. In the recent ten years, not only Diophantine equation itself develops very actively, but also promotes the development of other fields.

11. An elementary proof of the Diophantine equation (The equation abbreviated ) is given by ...

12. By elementary rank transformations of matrix, a method has been obtained to solve linear diophantine equation with some variables. The method overcomes the deficiency of traditional method.

13. Conventional predictive control algorithm possesses good overall performance, but it needs to solve the complicated Diophantine equation.

14. In this paper, by the elementary transformation of matrix and diophantine equation,[Sentencedict] two simple methods of solving linear congruence equation system are presented.

15. This paper is to study several special kinds of Diophantine equation which contains the following five aspects:1.

16. In this thesis the basic principle of the Generalized Predictive Control (GPC) algorithm is introduced. Based on this algorithm, a modification of the Diophantine equation recursion is developed.