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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.

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  • $\begingroup$ Yes by the Hilbert-Waring theorem. $\endgroup$ – Poon Levi Apr 6 '18 at 4:13
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This is Waring's problem and the answer is yes.

See https://en.wikipedia.org/wiki/Waring%27s_problem

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