For n >= 3, how close can a^n + b^n get to c^n, if a <= b < c ?
With n = 3, there are quite a few combinations where a^3 + b^3 = c^3 +/- 1. There is a parametric solution for a^3 + b^3 = c^3 - 2, with the first example 5^6 + 6^6 = 7^6 - 2. So there are a huge number of near misses.
For n = 4, it took my computer a very short time to find some values where a^4 + b^4 = c^4 when calculated with double precision floating point arithmetic, but the correctly calculated values where about a million apart.
Is there anything known for n >= 4? For example the obvious next question would be: Is it possible that a^n + b^n = c^n +/- 1 if n >= 4 and 1 <= a <= b < c?