正整數(shù)n的一個(gè)拆分是指將n表示為一個(gè)或多個(gè)正整數(shù)的無序和。n的不同拆分方式數(shù)稱為n的拆分?jǐn)?shù)。給出了一個(gè)正整數(shù)n能拆分成連續(xù)奇數(shù)和連續(xù)偶數(shù)之和的充要條件，并求出了這兩種拆分的拆分?jǐn)?shù)。將其結(jié)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(gè)數(shù)的確定。證明了如果n能表示成連續(xù)奇數(shù)和連續(xù)偶數(shù)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.