Let $p \geq 1$ be odd. Is there always an $M \in \mathbb{Z}_{>0}$ such that for any $n \in \mathbb{Z}$ there exist (not necessarily distinct) integers $a_1, a_2 \cdots a_{M}$ such that $n = \sum_{i=1}^{M} a_i^p$? In other words, if we fix an odd number $p$, is there a number $M$ such that any integer $n$ can be written as the sum of $M$ $p$-powers.
This question is motivated by (and a strengthening of) this fact, namely that any integer can be written as the sum of $8$ cubes.
