$m^k-k^m=17$ find solutions. Find all solution to the equation 
$m^k-k^m=17$  where $m,k\ge 2$ and  are positive integers.
I have analysed the equation by $mod(3) $  but I didn't get anything.

Let's consider when $k>m$. Let's $f(m,k)=m^k-k^m$.

*

*$k=m+1$
$g(m)=f(m,m+1)=m^{m+1}-(m+1)^m$,$g(2)=-1,g(3)=17,g(4)=1024-625>17,g(5)>17,...$


*$k=m+2$
$g(m)=f(m,m+2)=m^{m+2}-(m+2)^m$,$g(2)=0,g(3)>17,g(4)>17,g(5)>17,...$


*$k=m+3$
$g(m)=f(m,m+3)=m^{m+3}-(m+3)^m$,$g(2)=7,g(3)>17,g(4)>17,g(5)>17,...$


*$k=m+4$
$g(m)=f(m,m+4)=m^{m+4}-(m+4)^m$,$g(2)>17,g(3)>17,g(4)>17,g(5)>17,...$
.....
So I think for $k>m$ your solution the only solution.
