Let $p\geq 5$ be a prime number, let $E= \{(a,b,c,n) \in \mathbb{N}^4 ~|~ a^p+b^p+c^p=3^n\}$.


I know for all $k \geq 0$, $(3^k,0,0,kp), (0,3^k,0,kp), (0,0,3^k,kp) \in E$ and $(3^k,3^k,3^k,kp+1) \in E$.


Let $F\_1=\{(3^k,0,0,kp) ~| ~k \in \mathbb{N}\}$, $F\_2= \{(0,3^k,0,kp) ~| ~k \in \mathbb{N}\}$, $F\_3= \{(0,0,3^k,kp) ~| ~k \in \mathbb{N}\}$ and $F\_4=\{(3^k,3^k,3^k,kp+1) ~| ~k \in \mathbb{N}\}$.


Is $E=F\_1 \cup F\_2 \cup F\_3 \cup F\_4 $ ? Otherwise stated: 



> 
> **Are there non trivial solutions for $p\geq 5$ ?**
> 
> 
> 


If $p=2$ or $3$, there are other solutions. For example, for $p=3$, $1^3+6^3+8^3=3^6$. For $p=2$, $1^2+1^2+5^2=3^3$. 


I have asked the same question on MathStackExchange at this [link](https://math.stackexchange.com/questions/2171338/apbpcp-3n).


Thanks in advance.



Non trivial solutions of $a^p+b^p+c^p=3^n$
Partial answer. The $n$-conjecture implies at most finitely many
counterexamples over the *integers* for $p \ge 5$ besides yours
and some additional negative.


The [n-conjecture](http://cr.yp.to/bib/1994/browkin.pdf). 
is a generalization of abc and basically says that the if
$a\_1 + \ldots + a\_n=0$, no proper subsum vanishes and $a\_i$
are coprime, then the radical of $a\_1\cdots a\_n$ can't be
too small.


For coprime $3^n,a,b,c$, possibly after clearing the gcd, the radical
is at most $3abc$.


