Sum of powers in multiple ways

Question

I came across this little tidbit in a book with no reference or extra explanation. We have $$1^n+6^n+8^n=2^n+4^n+9^n\qquad \forall 1\leq n\leq 2,$$ $$1^n+5^n+8^n+12^n=2^n+3^n+10^n+11^n\qquad \forall 1\leq n\leq 3,$$ and $$1^n+5^n+8^n+12^n+18^n+19^n=2^n+3^n+9^n+13^n+16^n+20^n\qquad \forall 1\leq n\leq 4.$$

I have no idea how to search for this, and whatever I have tried has turned up empty. Does anyone know if this has a name of some kind? If this is a duplicate, I couldn't find it.

More specifically, and what I care about, can we continue this? In other words, for every $N$, do there exist distinct $a_1,\ldots,a_r,b_1,\ldots,b_r$ such that $\sum a_i^n=\sum b_i^n$ for all $1\leq n\leq N$?

Answer 1

Turned into an answer and added more details on request.

This type of equations is called Multigrade Equations, or more specifically, the Prouhet-Tarry-Escott Problem. (Links are to their respective MathWorld articles). Six of a kind . contains some more discussion of this problem.

As to whether a solution exists for each $N$, this answer contains a constructive theorem by Tarry-Escott which proves the affirmative, with $r = 2^N$. Hence the interest in the problem is to find ideal solutions, where $r = M+1$ as they are minimal.