If x and y have no common factor >1 then both factors on LHS are z-th powers which means x and y are multiples of z, say x=az, y=bz.
Substitute those values and take the z-th root to get (az)^a + (bz)^b = z which requires a=b=1, x=y=z=1. This is the solution given.
Let x=dp, y=dq, z=dr, gcf(p,q)=1.
Substitute and take the d-th root to get d^(p+q) * p^p * q^q = d^r * r^r.
The powers of d must be equal so factor them out leaving p^p * q^q = r^r which reverts to the first case.
So (x,y,z)=(1,1,1) is the only solution.
Posted by xdog
on 2014-07-13 10:40:04