Let be a fixed positive integer. Let

.

Let us consider expressions of the form

where .

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

.

Thus:

**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.