Research on Intuitive Solution and Uniqueness of the Tower Exponent Equation

Abstract

Taking pure elementary algebra as the research tool, this paper solves the three-layer right-associative tower exponent equation without advanced knowledge such as logarithms and calculus throughout the whole process. First, integer solutions are strictly eliminated via integer recursive scaling, zero solutions are excluded by analyzing domain boundaries, and combined with the growth law of power operations, it is predicted that the solution is in the form of positive real numbers with fractional exponents. Based on the expression form of positive real powers, variable substitution and derivation are directly carried out to obtain the particular solution , which is verified by substitution. Rigorous demonstrations are conducted on negative real solutions and interval monotonicity: it is directly proved that the equation has no solution when , and strict monotonicity is proved by definition only in the interval , completing the proof of uniqueness of real solutions. Meanwhile, the definitional contradictions of negative base numbers in the real number field are clarified, negating the existence of negative real solutions. This paper retains the intuitive and original reasoning logic, revises the connection defects of argumentation, and forms a complete, rigorous and self-consistent research compatible with the elementary mathematics system. 本文以纯初等代数方法为工具,求解三层右结合指数塔方程 ,全程不使用对数、微积分等高阶知识。首先通过整数递推放缩排除整数解,补充定义域边界排除零解,结合幂运算增长规律预判解为正实数分数指数形式;基于正实数幂的表达形式直接设元推导,求得特解 并代入验证。针对负实数解、区间单调性等问题展开严谨论证:直接证明 时方程无解,仅在 区间用定义法证明严格单调性,完成实数解唯一性证明;同时厘清负数底数的实数域定义矛盾,否定负实数解存在性。全文保留直观原生推理逻辑,修正论证衔接问题,形成严谨、自洽、适配初等数学体系的完整研究。