Positive integral solutions $(m,n)$ to $m = n^m$

Question

Are there any positive integral solutions $(m,n)$ to the diophatine equation $n=m^{n}$ besides $(m,n)=(1,1)$? Not sure how to approach this question. I got the (obvious) solution by guessing. It seems clear that $m\leq{n}$.

Answer 1

No,there are no other positive integral solutions except the one you have already guessed. For this solution ,I would be replacing (n,m) with (x,y).(as I comfortable in solving with those variables)

$\implies$$x=y^{x}$

$\implies y=x^{1/x}$

By differentiating the expression,you will realise that the maximum value of the expression is at $e$ which is $e^{1/e}$. You can prove that $e^{1/e}<2$ since $\frac{1}{e} (≈0.37)< ln(2) {≈0.69}$.

As,you can observe that the only integral value that y can be is 1.Thus the only positive integral solutions is (1,1).

Answer 2

$$m=1^m$$ has the only (and obvious) solution $m=1$.

There are no other solutions to the initial problem, as

$$m<2^m<3^m<\cdots$$

[By induction, $1<2^1$ and $m<2^m\implies m+1<2m<2^{m+1}$.]