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](http://www.math.lviv.ua/szkocka/viewpage.php?vol=2&page=95) of [Volume 2](http://www.math.lviv.ua/szkocka/viewbook.php?vol=2) of the [Lviv Scottish Book](http://www.math.lviv.ua/szkocka).


**The prize:** A personal congratulation :)



Representing natural numbers as sums of powers of distinct numbers
One can represent N+1 as N^1 + 1^N. There does not seem to be a representation of 2 without using 0 and three or more terms. 1 (and some other powers) needs only one term.


The problem needs major alteration to be suitable for this forum.


Gerhard "More Simple Thinking Was Needed" Paseman, 2019.08.23.


