Tell HN: Elementary Proof of 2,4 being the only non trivial solution to x^y=y^x

Context: https://news.ycombinator.com/item?id=35635491

Without loss of generality, assume a > b [ Or the otherway for that matter, because a != b ]

Rewriting the equation we get (a/b)^b = b^(a-b)

Right side is integer raised to integer and thus an integer and left side is fraction raised to integer. So that fraction is not really a fraction.

In other words a is a multiple of b

=> a = c * b where c is an integer > 1 [ Because a != b ]

=> c^b = b^(b * ( c - 1)) [ Substituting a with c*b in the original equation ]

=> c = b^(c-1) [ Taking bth root on both sides ]

c != 1 [ Because a != b ]

c != 3 [ Because b^2 = 3 has no integer solutions ]

c != 4 [ Because b^3 = 4 has no integer solutions ]

And so on..

=> c = 2 is the only possible solution

=> Substituting c with 2 we get b = 2 and a = 4

Edits: Formatting and Explanation