实数n次勾股定理
The Real Number n-Degree Pythagorean Theorem
摘要：
本文将勾股定理推广到正实指数和负实指数，使其具有a^n + b^n = c^n的形式，并利用定义gamma = b/a >= 1。对于正实数集合中的n的情况，顶角为实数需要n大于或等于1，并且对伽玛没有超出其定义的限制。然而，对于负实数集合中的n，本工作发现了a^n + b^n = c^n产生实顶角所必需的两个重要限制：1 <= gamma < 2，并且n不能超过依赖于gamma的临界值。此外，已经确定了相关三角形的面积以及这些面积为最大值或最小值的条件。
英文摘要：
This paper extends the Pythagorean Theorem to positive and negative real exponents to take the form a^n + b^n = c^n and makes use of the definition gamma = b/a >= 1. For the case of n in the set of positive real numbers, n greater than or equal to 1 is necessary for the vertex angle to be real, and there are no restrictions on gamma beyond its definition. However, for n in the set of negative reals, two significant restrictions that are necessary for a^n + b^n = c^n to yield real vertex angles have been discovered by this work: 1 <= gamma < 2, and n cannot exceed a critical value which is gamma-dependent. Additionally, the areas of the associated triangles have been determined as well as the conditions for those areas to be maxima or minima.
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