The generalized Fermat conjecture

If a, b, c are non-zero integers, we considerer the following problem: for which values of n the line ax + by + cz = 0 may be tangent to the curve x n + y n = z n ? We give a partial solution: if n = 5 or if n-1 is a prime a number, then the answer is the line cannot be tangent to the curve. This pr...