21 problems

For each $k\in\mathbb{N}$, let $g(k)$ be the smallest number such that every positive integer is a sum of $g(k)$ non-negative $k$-th powers. Waring's conjectural formula. It is con…

Let $C$ be the classical Cantor ternary set, and let $x_1,x_2,x_3,x_4$ be elements of $C$. Athreya–Reznick–Tyson's four-squares conjecture. Every element of $[0,1]$ can be expresse…

Higher-dimensional Waring-type conjecture. If $d$ is fixed and $k$ is sufficiently large, then the equation holds only if

Let $b_i=i^k$ and consider the equation … A solution is trivial when the two multisets of summands agree, namely when the right-hand indices are a permutation of the left-hand indi…

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…

Let $m>2$ be an even integer, and let $g(m)$ denote the least positive integer such that every positive integer is expressible as the sum of at most $g(m)$ $m$th powers. Vaughan–Wo…

Larsen's conjecture. For a given $k$, there exists a constant $N(k)\in\mathbb N$ depending only on $k$ such that, for all pairs $(n,q)$, if

Four-cubes conjecture. One has

Waring-number conjecture.

A sufficiently large integer means an integer exceeding some fixed bound. Four-cubes conjecture. It is conjectured that every sufficiently large integer is a sum of four non-negati…

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…

For a positive integer $n$, let $r_k(n)$ denote the number of representations of $n$ as a sum of $k$ positive integral $k$-th powers, and let $\mathscr{C}$ be the set of integers r…

Linear-summand conjecture. $(W,s)$ is a Waring pair for some $s=O(k)$.

For each positive integer $k$, let $g(k)$ denote the least integer such that every positive integer is a sum of $g(k)$ $k$th powers. Hilbert–Waring conjecture. It is conjectured th…

For integers $n,k\geq 1$, define $C_n$ to be the set of $kn$-bit binary $k$th powers … so that $C_n$ has cardinality $2^{n-1}$. For sets $S$ and $T$, their sumset is…

A natural number is a binary cube if it is a power of two raised to the third power. Nine binary cubes conjecture. Every natural number greater than $147615$ is the sum of at most…

Let $k\geq 3$ be an integer, let $d$ be the dimension, and let $A_*$ denote the corresponding discrete maximal operator on $\ell^p(\mathbb{Z}^d)$. Write $\widetilde{G}_1(k)$ for th…

Let $\tilde{G}(k)$ be the smallest integer $s$ for which the circle-method asymptotic formula for the number $r_{s,k}(l)$ of representations of $l$ as a sum of $s$ positive $k$th p…

Paucity conjecture. If $2\leq s<k$, the diagonal solutions dominate, so that

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 $k\ge 2$,