The Real Number n-Degree Pythagorean Theorem

TL;DR

This paper generalizes the Pythagorean theorem to real exponents, exploring conditions for real vertex angles and triangle areas, revealing new restrictions for negative exponents and deriving geometric properties.

Contribution

It introduces conditions for real vertex angles in the real number n-degree Pythagorean theorem, including restrictions on gamma and exponent bounds, and analyzes triangle areas.

Findings

For positive n >= 1, vertex angles are real without gamma restrictions.

For negative n, gamma must satisfy 1 <= gamma < 2, with an upper bound depending on gamma.

Triangle areas are characterized, with conditions for maxima and minima.

Abstract

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.