All solution of some equation

Question

Let $A=\{(m,n)\in\mathbb{N\times N}:m\neq n \text{ and } m^n=n^m\}$. It is clear that $(2,4),(4,2)\in A$. What is the solution of this equation ?

Answer 1

HINT : $$m^n=n^m\iff n\ln m=m\ln n\iff \frac{\ln m}{m}=\frac{\ln n}{n}.$$ Then, consider the graph of $\frac{\ln x}{x}$.

Answer 2

For purely algebraic proof:

Suppose $m\ge n.$ Note that should there exist a positive integer $k$ such that $$m=n^k.$$ Then $$m^n=n^{kn}=n^m.$$ That is $$m=kn.$$ $$kn=n^k$$ $$k=n^{k-1}.$$ If $k=1,$ it gives $n=m.$
If $k=2,$ it gives $n=2, m=4$
If $k\ge3$ and $n\ge2,$ then $n^{k-1}>k.$
(By mathematical induction we can show that $2^{k-1}>k$ for $k\ge3.$ )
Hence the only solution with $n\not=m$ is $n=2, m=4.$