A conjecture relate to Euler's sum of powers conjecture

Question

I am ZHICHEN LIU(Will). I made a conjecture based on Euler's sum of powers conjecture. I think it works, but I don't know if anyone made this conjecture before. So I hope anyone can give me some directions, or give to me one counterexample to this conjecture I made.

Euler's conjecture: Assume $X_1$, $X_2$,$X_3$, ... , $X_n$ are integers, and m is positive whole number.

$X_1^m+X_2^m+X_3^m+...+X_n^m=Y^m$ (1)

If equation (1) holds, then Y won't have integer solution if m is greater than n.

My conjecture (My apologize if someone did this before.):

Assume $X_1$, $X_2$,$X_3$, ... , $X_n$ are integers, additionally, $X_1$, $X_2$,$X_3$, ... , $X_n$ are relatively prime pairs , and m is positive whole number.

$X_1^m+X_2^m+X_3^m+...+X_n^m=Y^m$ (2)

then If equation (2) holds, then Y won't have integer solution if m is greater than n.

(Basically, I think the Euler's sum of powers conjecture need $X_1$, $X_2$,$X_3$, ... , $X_n$ to be relatively prime pairs)

For example:

$X_1$, $X_2$,$X_3$ are integers,

Additionally, $X_1$ and $X_2$ are relatively primes, $X_1$ and $X_3$ are relatively primes, $X_2$ and $X_3$ are relatively primes.

$X_1^3+X_2^3+X_3^3=Y^3$ (3)

then if equation (3) holds, then Y won't have integer solution.

Answer 1

Counterexample: $$55^5 + 3183^5 + 28969^5 + 85282^5 = 85359^5$$

Answer 2

See https://sites.google.com/site/sevensixthpowers/ for (6,1,7) solutions, showing examples where many $X_i$ must share one or more common factors, due to modular constrains (mod 7, 8 and 9 in this case).