奇数集合的一个二元分解:由二进制最低位指数导出的两类 显式公式 A Binary Decomposition of the Odd Integers via the Parity of the Lowest 2-adic Exponent

本文研究由参数 m,t∈N+ 定义的两个代数式,并证明它们生成的奇数集合互不重叠,且并集恰为全体奇数。证明基于奇数的二进制结构,特别是最低位 2-进制指数(2-adic 赋值)的奇偶性,从而得到奇数集合的一种自然分类。该结果为奇数提供了一种结构化的分解方式,并与整数序列、同余结构及离散动力系统等方向具有潜在联系。 A structural decomposition of the odd integers is established through two explicit algebraic formulas parameterized by positive integers m and t. The resulting two sets of odd integers are shown to be disjoint and their union equals the full set of odd integers. The proof relies on the binary expansion of integers and, in particular, on the parity of the lowest 2-adic exponent. This yields a natural classification of all odd integers into two families determined by their 2-adic structure. The work provides an elementary yet structurally meaningful perspective on odd integers, with potential connections to integer sequences, modular arithmetic, and discrete dynamical systems.