Show that $17$ is the only prime of the form $p^q +q^ p$ , where $p$ and $q$ are prime 
Show that $17$ is the only prime of the form $p^q +q^ p$ , where $p$ and $q$ are prime

My attempt so far is first assume $p$ and $q$ are prime. Now $17=2^3+3^2.$
Now fix $p=2$ and let $q>3$ then $q=3x+1$ or $q=3x+2$, $x \in \mathbb{Z}$
then $2^{3x+1}+(3x+1)^2=9x^2+6x+2(2^{3x})+1$
Now I'm not really sure what to do from here my goal was to show its not prime from both cases of $q$ but I think I"m missing the algebra necessary to show that. 

If $p,q$ are both odd (or both even) then $p^q+q^p$ is even and $>2$, hence nt prime.
So if $n=p^q+q^p$ is prime we can assume wlog that $q=2$ and $n=p^2+2^p$ with $p$ odd.
Then $2^p\equiv 2\pmod 3$.But if $p$ is not a multiple of $3$, then $p^2\equiv 1\pmod 3$ and so $2^p+p^2$ is a multiple of $3$. As $2^p+p^2$ cannot be equal to $3$,  the only valid remaining case is $p=3$, $n=2^3+3^2=17$.
