A Conjecture on the Fermat Function

For any positive integer n let F(n) denote the smallest absolute 
value of  x^n + y^n + z^n  for non-zero integers x,y,z WITH DISTINCT 
ABSOLUTE VALUES.  The first few  values are

                    n        F(n)
                   ---      ------
                    1          0
                    2         14
                    3          1
                    4         98
                    5         12 (?)
 
Obviously for even values of n we have F(n) = 1 + 2^n + 3^n, which is 
not very interesting.  For odd values of n the determination of F(n) 
seems non-trivial, considering that it is very difficult to say if F(n) 
ever equals zero for any n>1.  I'm not even sure that F(5)=12, but 
certainly it is no greater than 12 (in view of 13^5 + 16^5 + (-17)^5).  
Can anyone fill in some more values of F(n)? 

We could also define F(n) as the smallest absolute value of  
x^n + y^n - z^n  where x,y,z are distinct positive integers.  
This gives an interesting function for even values of n as
well.

An interesting conjecture along these lines is that there is no 
solution in distinct positive integers x,y,z to the inequality  

          |x^n + y^n - z^n|   <   3^n - 2^n - 1  

for n > 5.