正整数的连续奇偶拆分

正整数n的一个拆分是指将n表示为一个或多个正整数的无序和。n的不同拆分方式数称为n的拆分数。给出了一个正整数n能拆分成连续奇数和连续偶数之和的充要条件，并求出了这两种拆分的拆分数。将其结果用于讨论不定方程x2−y2=n，给出了判断该方程解的存在性条件，以及解的个数的确定。证明了如果n能表示成连续奇数和连续偶数之和，则表示法唯一。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.