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

a quick guess is that a), b) and d) are never possible and c) is always possible.

an irrational number should always exist:
If x=2 then y^y = 5 and y is a bit over 2
If x=3 then y^y = 36 and y is a bit over 3
the real question is whether any of these is rational.  I don't think raising a rational non-integer to itself can yield an integer you might not even be able to make it rational.

for example (9/2)^(9/2) = 9^(9/2) / 2^(9/2) = 19683 / 16√2

as for part d) a complex to a complex power is not well defined, but I don't think it can give a real value if the base and power are the same.  I may have to play around with this one.

Posted by Jer on 2012-01-22 02:00:19