Please reference prior post: WHY DOES the PYTHAGOREAN THEOREM WORK?
For any triangle ABC where c* is the long leg, an exponent n exists that will satisfy a n + b n = c n of non-Diophantine concern.
*Alternatively, a n + b n = c n can be re-written b n – a n = c n at the isosceles inversion point to maintain constancy of convention. This is necessitated as ∠C changes between the degenerate triangle limits of zero and π radians, the transition from obtuse scalene to acute scalene. Constrained by this equation, as triangles of a : b < 1 approach isosceles, n becomes extreme. In the case of a : b = 1, the equation is invalid at the equilateral where c fails to be the long leg.
