正整数的连续奇偶拆分

Partitions of Positive Integer as the Sum of Continuous Even or Odd Numbers

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 x^2-y^2=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 rn continuous even or odd numbers, then the representation is unique.