Some Determination Methods of Elementary Numbers and Divisibility Relationships

本文探讨了初等数的质合数判定方法和整除关系，通过数学证明展示了特定函数形式的合数集、特定数字累加和的整除性质，以及质数与合数的图形规律。例如定理1、2所确定的函数形式的合数集，在初等数的质合数判别起着判定的作用，又如定理3和4、5的整除关系与寻找数形上的结合规律的证明过程中，以奇偶性质推理出的逻辑是整除关系的重要证明方法之一。所以本文工作内容为：探究和寻找一些以函数形式存在的合数集，特殊数字的n次方累加和能被特定数字整除，以数形结合探讨质合数的规律。This paper explores methods for distinguishing prime and composite numbers in elementary numbers and their divisibility relationships. Through mathematical proofs, it demonstrates the specific functional forms of composite number sets, the divisibility properties of the cumulative sums of specific numbers raised to powers, and the graphical patterns of prime and composite numbers. For example, the composite number sets defined by Theorems 1 and 2 play a crucial role in the distinction of prime and composite numbers in elementary numbers. Additionally, in the process of proving the divisibility relationships in Theorems 3, 4, and 5 and seeking the combined patterns of numbers and shapes, logical reasoning based on the properties of odd and even numbers is one of the important proof methods for divisibility relationships. Therefore, the work of this paper includes exploring and identifying composite number sets that exist in functional forms, investigating whether the cumulative sum of the n-th powers of special numbers can be divisible by specific numbers, and exploring the patterns of prime and composite numbers through the combination of numbers and shapes.