Why does $223$ require $37$ terms to be expressed as a sum of positive fifth powers?

Question

I saw this post about the number earlier today with this formula.

$a_1^5 + a_2^5 + \cdots + a_k^5$, if $k < 37$

Why is it that $223$ is the only number to have to be $37$ or up to be represented by powers of five?

Answer 1

Waring's Problem states that, given $k$, every number can be represented as the sum of at most $s$ $k$-th powers, i.e. that the number of $k$-th powers you need does not keep growing for larger numbers. Apparently this was proven by David Hilbert.

Given that there is a bound, there must be at least one number that is is the worst case. For $5$th powers the bound happens to be $37$, and there happens to be only one number that attains that bound which is $223=6\cdot2^5+31\cdot1^5$.

$3^5=243=7\cdot2^5+19\cdot1^5$, so you won't need more than $37$ fifth powers for any other number below that. For larger numbers it gets trickier, because the "greedy" algorithm where you use as many large fifth powers as possible does not always work.

For example, the greedy algorithm would give $3000 = 2\cdot4^5 + 3\cdot3^5 +6\cdot2^5 + 31\cdot1^5$, using $42$ fifth powers, but by breaking up one of the $4^5$ terms you get $3000 = 1\cdot4^5 + 8\cdot3^5 +1\cdot2^5$ which uses only $10$ fifth powers.

For larger numbers there is much more flexibility in the representations allowing you to keep the number of powers below $37$. Actually proving that there is always enough flexibility to do this is difficult.