Let be a fixed positive integer. Let
Let us consider expressions of the form
Note that takes possible values in which are all congruent modulo . Hence the number of distinct values that takes is at most
Since this is , it is eventually dominated by . Thus for all sufficiently large , we can find such that
with for some .
Note that . Let and . Then and are non-empty and disjoint, , and
Proposition. Let be a fixed positive integer. Then one can find distinct positive integers such that
In particular, one can choose .
Example. Let . Note that for all . So there exist distinct integers such that . In particular, one has
as well as
Likewise, for fifth powers we can find such ‘s and ‘s in .
This wikipedia article contains many results related to sums of powers.