Previously I had asked a question about a Diophantine equation linked here.
I have come back to think about this question but in a different manner.
So here is the set up:
Let A and $a_i$ be integers such that $A \ge a_i \ge 0$ for every i, and A is non-zero.
The sum in question now is,
$\sum_{i=1}^ka_i^k=A^k$
Dividing by A^k,
$\sum_{i=1}^k({a_i\over A})^k=1$
Now we ask for what sequences of $a_i$ is this true for $k \ge 3$?
What if take the lim of this?
$\lim_{k \to \infty}\sum_{i=1}^k({a_i\over A})^k $ and ask if it converges to 1?
Yes, I know that we can rewrite the ${a_i \over A}$ as $b_i$ to make the sum: $\sum_{i=1}^kb_i^k=1$, and the limit $\lim_{k \to \infty}\sum_{i=1}^kb_i^k $, but I would rather not lose any information pertaining to the sequence.
My question is what now? Have I made this too general to solve, prove, or this disprove? Has anyone else made this approach? If I am close what tools am I going to need to show either? What restrictions might be better to change?
I had thought about this approach after thinking about the connection between compounding interest and continual interest.
For particular $k$ you might get particular integer solutions. It would only make sense to take a limit if you had a family of solutions parametrized by $k$, where $a_i = f(i,k)$ for some function $f$ and $A = A(k)$. I don't know if there are such families of solutions
EDIT: (other that the trivial $A^k + 0^k + \ldots + 0^k = A^k$).