Only one pair of distinct positive integers satisfy the equation m^n = n^m
I am struggling to see the great revelation.<p>I am not a mathematician, but in the discrete domain of integers:<p>1) you have two functions, essentially we look for where the two 3d space surfaces intersect, if z is taken as the result for each equation.<p>2) the integers &quot;are disctinct&quot;, so (0,0) and (1,1) are out, plus (2,2) (3,3) etc. Basically a whole linear diagonal in the instersection of both 3d spaces is excluded (why though, to what useful end?)<p>3) Starting points for the ranges is therefore (0,1) and (1,0)<p>4) 1^y is always 1, and x^0 is always 1 so there is a constant starting value of 1 both on both axis<p>5) but x^y will always be larger than y^x, for y&gt;x and x^y &gt; y^x 
(prove this by taking the first derivative to get rate of change. Do you use Laplace z domain for discrete, instead of s for continuous?)<p>6) and the converse to 5.<p>So once you have found one solution, you know to stop looking, the two surfaces keep diverging from each other.<p>Why resort to the continuous domain to solve a problem n the discrete, is this even a valid approach?<p>eg Is there a formal proof that says integers strictly follow that same rules as the continuous domain, just as a subset? I&#x27;m interested.<p>Does this come under group theory, a set with an applied operation?<p>As I said I am not a mathematician, I am an electrical engineer so probably one of the worst abusers of pure math in a formal sense, but the more I think about this the more questions this raises in my thinking.<p>Can someone point out errors in thinking?