正整數的連續(xù)奇偶拆分

正整數n的一個拆分是指將n表示為一個或多個正整數的無序和。n的不同拆分方式數稱為n的拆分數。給出了一個正整數n能拆分成連續(xù)奇數和連續(xù)偶數之和的充要條件，并求出了這兩種拆分的拆分數。將其結果用于討論不定方程x2?y2=n，給出了判斷該方程解的存在性條件，以及解的個數的確定。證明了如果n能表示成連續(xù)奇數和連續(xù)偶數之和，則表示法唯一。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.