The equation 1^x+2^x+3^x+4^x+5^x+ ... x^x=y^y has a trivial solution : x=y=1
Prove the uniqueness of this solution or provide additional solution(s) if x is a positive integer and y is:

a) integer as well
b) a fractional number
c) an irrational number
d) a complex (non-real) number