Waring's Problem: Exact Values

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $k\in\mathbb N$, and let $\mathbb Z_{\ge0}=\{0,1,2,\ldots\}$. Define $R_{k,s}(n)$ to mean that there exist $x_1,\ldots,x_s\in\mathbb Z_{\ge0}$ with $n=x_1^k+\cdots+x_s^k$. Zero summands permit representations using at most $s$ positive powers. Set \[g(k)=\min\{s\in\mathbb N:\forall n\in\mathbb N, R_{k,s}(n)\},\] \[G(k)=\min\{s\in\mathbb N:\exists N\in\mathbb N\ \forall n\ge N, R_{k,s}(n)\}.\] Thus $g$ requires every positive integer, while $G$ allows finitely many exceptions. Hilbert's theorem ensures these minima exist for $k\ge2$, and $g(1)=G(1)=1$.

Statement recorded in the supplied source.

Determine the exact integers $g(k)$ and $G(k)$ for every integer exponent $k\ge2$ (with the trivial $k=1$ values as above). Both the universal and sufficiently-large representation functions are included. [S2] [S3] The two functions have different quantifiers: determining a formula for $g(k)$ alone does not determine $G(k)$.

Short English statement

Determine exactly how many nonnegative integral kth powers suffice for every positive integer and how many suffice after excluding finitely many small integers, for each exponent.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1269 (NT-062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440 (version dated January 24, 2004), Section 2.2, pp. 16--17, discussion of Waring's problem. https://arxiv.org/pdf/math/0312440.
• S3 Jörg Brüdern and Trevor D. Wooley, On Waring's problem for larger powers, arXiv:2211.10380v1 (2022), Section 1, p. 1. https://arxiv.org/pdf/2211.10380v1.