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.

Thanks in advance.


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. 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 $3abc$.


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