the only perfect number of the form (x to the power of n) + (y to the power of n)

Question

Perfect numbers have the form
2^k−1(2^k − 1)
The first few perfect number are 6,28,496 and 8,128.
Does n have to be something other than 1 in your question?
I can see 28 = 1³ + 3³
but I'm not sure it's the only number that can be written like this, at least I've never seen a proof or conjecture on this.

sarah, do you go to OCHS?