In order for x^y = y^z = z^x to hold true, it must mean that: x = y = z. Otherwise, if they are not equal to each other, x^y will never equal to y^z or to z^x. Therefore, x must equal y and must equal z. Or, x = y.
Just to confirm the above, here is a short computer code that spits out EQUAL values for x, y and z, no matter what initial values you assign to them.
a=1; b=1;d=1;c=a^b; e=b^d;f=d^a;if(c==e and e==f, goto7, goto10);print"x=", a,", ",; print"y=",b,", ",; print"z=",d,", ",; a++;if(a<100, goto3, 0);a=1;b++;if(b<100, goto3, discard=0; a=1;b=1;d++;if(d<100, goto3, 0)
x= 1 , y= 1 , z= 1 , x= 2 , y= 2 , z= 2 , x= 3 , y= 3 , z= 3 , x= 4 , y= 4 , z= 4 , x= 5 , y= 5 , z= 5 , x= 6 , y= 6 , z= 6 , x= 7 , y= 7 , z= 7 , x= 8 , y= 8 , z= 8 , x= 9 , y= 9 , z= 9 , x= 10 , y= 10 , z= 10 .............and so on.
