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 problem is strongly related to Fermat' s Last Theorem. (C) 2019 Mathematical Institute Slovak Academy of Sciences