Viewing profile — nmmnthrowaway

Yes, but there is a much simpler proof.

That is correct. Why are there no others?

I think a more interesting problem to consider after solving n^m = m^n is solving n^m = m^n + 1 in the natural numbers.

n^n * n^(m - n) = n ^ m = m ^ n, so n^n divides m^n, so n divides m.