# Representing natural numbers as sums of powers of distinct numbers

Find the smallest number $$n$$ such that almost all natural numbers can be represented as the sum $$a_1^{a_{p(1)}}+a_2^{a_{p(2)}}+\dots+a_n^{a_{p(n)}}$$where $$a_1,\dots,a_n$$ are pairwise distinct natural numbers and $$p$$ is a permutation of the set $$\{1,\dots,n\}$$.

The problem was posed on 24.03.2019 by Jacek Jurewicz on page 95 of Volume 2 of the Lviv Scottish Book.

The prize: A personal congratulation :)

• N=2. This works for numbers greater than N. (Hint: a_1=1.) Gerhard "You Can Congratulate Me Here" Paseman, 2019.08.23. Aug 23, 2019 at 17:11
• @GerhardPaseman Could you please write more details of your solution, desirably as an answer? Thank you. Aug 23, 2019 at 17:14