23 is Largest Integer not Sum of Distinct Perfect Powers

Theorem
The largest integer which cannot be expressed as the sum of distinct perfect powers is $23$.

Proof
The first few perfect powers are:
 * $1, 4, 8, 9, 16, 25, 27, 32, \dots$

First we show that $23$ cannot be expressed as the sum of distinct perfect powers.

Only $1, 4, 8, 9, 16$ are perfect powers less than $23$.

Suppose $23$ can be so expressed.

Since $1 + 4 + 8 + 9 = 22 < 23$, $16$ must be used in the sum.

However $23 - 16 = 7$ cannot be expressed as a sum of $1$ and $4$.

Thus $23$ cannot be expressed as the sum of distinct perfect powers.

Now we show that all numbers greater than $23$ can be so expressed.

By Richert's Theorem, we just need to show:
 * For any $23 < n \le 23 + 32$, $n$ can be expressed as a sum of distinct elements in $\set {1, 4, 8, 9, 16, 25, 27}$


 * $s_{i + 1} \le 2 s_i$ for every $i \ge 7$, where $s_i$ is the $i$th perfect power

Verification of the first statement is included in the bottom of this proof.

To verify the second statement:

Let $i \ge 7$.

Let $m$ be the integer satisfying:
 * $2^{m + 1} > s_i \ge 2^m$

We have that $m \ge 4$.

Hence $2^{m + 1}$ is also a perfect power.

There must be a perfect power greater than $s_i$ but not greater than $2^{m + 1}$.

Thus:
 * $2 s_i \ge 2^{m + 1} \ge s_{i + 1}$

Therefore $23$ is the largest integer that cannot be expressed as the sum of distinct perfect powers..

Here is $23 < n \le 55$ expressed as a sum of distinct elements in $\set {1, 4, 8, 9, 16, 25, 27}$: