Problems & Puzzles: Collection 20th Coll.20th-004. Puzzles 104 & 385, revisited.

On March 12, 2018, Seiji Tomita posed the following puzzle:

I suggest generalizing the puzzle 104 and 385 as follows.
N = x1^n + x2^n +....+ xn^n,n>3.
N=x1&x2&...xn
N: Prime number

Each number is sum of nth power of its n sections where
the sections for each number can have distinct lengths.
For example, ABCD = A^4+B^4+C^4+D^4, ABCDE = A^5+B^5+C^5+D^5+E^5.
2230433 = 22^4 + 30^4 + 4^4 + 33^4

Q1: Find large solutions for each nth power.
Q2: If possible, find the parametric solutions for each nth power.

Example for n=3, ((10^s-1)/3)^3+((2*10^s+1)/3)^3+1^3 = (10^s-1)/3*(10^s)^2+(2*10^s+1)/3*10^s+1.

Contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

Here is my only result :

45372981676573==45^7+37^7+29^7+81^7+67^7+65^7+73^7

There are a few other numbers that satisfy the first two conditions of the problem (for N= 5, 6 and 7)
but they are not prime.

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