TITLE: The equation $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.
QUESTION [5 upvotes]: I’m looking for positive integer solutions to $x^n+y^n+z^n=u^n+v^n+w^n=p$, where $p$ is prime.
Background.
I was looking at “Primes which are the sum of three nonzero 8th powers” https://oeis.org/A283019 and the like, and wondered if there are rules similar to those for a prime being the sum of two squares.
My efforts.
I’ve found primes of the form $x^n+y^n+z^n$ up to $n=19$. 
For $n=1$ solutions are trivial, for $n=2$ they are abundant. There are plenty of results for $n=3$ and they’re not uncommon for $n=4$.
However, for $\color{blue}{n=5}$ I’ve found just two lonely solutions.
$$(n,x,y,z=u,v,w,p)$$
$$(2,3,4,4=1,2,6,41)$$
$$(2,3,5,5=1,3,7,59)$$
$$(2,3,5,7=1,1,9,83)$$
$$(3,1,5,5=2,3,6,251)$$
$$(3,4,6,9=1,2,10,1009)$$
$$(3,1,9,9=4,4,11,1459)$$
$$(4,9,16,16=8,13,18,137633)$$
$$(4,4,18,19=1,6,22,235553)$$
$$(4,8,16,21=6,13,22,264113)$$
$$(\color{blue}5,11,183,209=19,168,216,604015282243)$$
$$(\color{blue}5,481,782,788=321,772,808,622015202536001)$$
My question.
Can anyone find more solutions for $n=5$, or $n>5$, or give any insights, please.

REPLY [3 votes]: I just have a look at the primes solutions on my website Equal Sums of Like Powers, and find a very interesting one for your question.
$$19+179+241=43+139+257=439$$
$$19^3+179^3+241^3=43^3+139^3+257^3=19739719$$
All the numbers, including $x=19,y=179,z=241,u=43,v=139,w=257,n=3,p=19739719,x+y+z=439$ are Primes!!!
BTW, I also find the following one
$$23+59+149+157+211=17+79+113+191+199=599$$
$$23^2+59^2+149^2+157^2+211^2=17^2+79^2+113^2+191^2+199^2$$
$$23^3+59^3+149^3+157^3+211^3=17^3+79^3+113^3+191^3+199^3$$
$$23^5+59^5+149^5+157^5+211^5=17^5+79^5+113^5+191^5+199^5=587777330999$$
All the items $(23,59,149,157,211,17,79,113,191,199)$ and the sums of each side ($599$), the sums of 5th powers ($587777330999$) are primes.