Positive Integer Spin-off Applications

A partition of a positive integer n is representation of n as an unordered sum of one or more positive integers. The number of different partitions of the positive integer n is called the partition number of n. There is important sense in theory and practice to study a partition of a positive integer n. The partition of the positive integer is an important issue in Combinatory, Graph theory and Number theory. G. W. Leibniz is the first mathematician to study it, and then Euler expanded it to the complete theory of partitions.In this thesis, we do the following work.Firstly, we do some discussion about the perfect partition. This paper gives an necessary and sufficient condition, an evaluation for the bound of part of partition and number of partition. Then we use the recursive method and have given two recurrent formulas of the perfect partition number with restrained condition of part of partition and number of partition. Moreover, we also have discussed about the generating function of perfect partition, and got some results of it.Secondly, we study the partition formula of three partition and its application. We introduce several proving methods of three partition, and give a new method different from the literature known to us that relatively simple formula to prove the partition formula of three partition. Then we discuss its practical application in triangle with integer sides,θgraph and molecule structure.Finaly, we discuss the partition of the positive integer and the number for positive integral solution of Diophantine equation. With the result known to us and the relation between p(n,k) and A(n,k), we give equivalent expression of p(n,k) when k=4, 5, 6 by resolving linear equation.