5 problems

Let $k$ be an even positive integer and let $e\geq 1$. Write $g_{2,e}(k)$ for the least number of $k$-th powers needed to represent every element of the ramified $2$-adic ring of r…

Waring-number conjecture.

Let $k\in\mathbb{N}$. For each positive integer-valued function $C:\mathbb{N}\rightarrow\mathbb{N}$, let $P_C(k)=\sup_{l\geq 2}\{S_C(k,l)\}$, where $S_C(k,l)$ is the minimum number…

Let $r_{k,k}(l)$ denote the number of representations of a positive integer $l$ as a sum of $k$ positive $k$th powers. Hooley's Hypothesis $K^*$. For $k=2$ and every $\epsilon>0$,…

Ideal Waring conjecture. For every $k\ge 2$,