Abstract: 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. In this paper, a sufficient and necessary condition of the positive integer n which can be represented as a sum of some continuous even or odd numbers is given. The partition numbers of these two kinds of partitions are also obtained. These consequences are used for research the equation x2-y2=n. The condition of the equation existence solution and number of solution are given. For given n and m, we also show that if n can be represented as a sum of m continuous even or odd numbers, then the representation is unique.