Solutions to the equation $xk = x^k$

Question

The equation, $xk = x^k$ (where $x$ and $k$ are both integers).

Are there any solutions other than $\{ (1,1), (2,2) \}$ ?

Answer 1

Well, $x^k=kx$ implies $x(x^{k-1}-k)=0$. Thus, $x=0$ works for all $k>0$ and $k=1$ works for all $x$. Aside from that, there are only solutions when $k$ has a $(k-1)$-th root, which I believe only occurs when $k=1$ or $k=2$.

Answer 2

There are no solutions unless $k=1$ or $x=k=2$. If $k=2$, $x^2=x*x \gt kx$ unless $x=2$. If $k \gt 2, 2^k \gt 2k$ as $2^k=(1+1)^k\gt 1+k+\frac 12k^2 \gt 2k$ and higher $x$ makes it worse.