6 problems

Let $H_3(n)$ denote the quantity used in the paper for the size or counting function of three-dimensional Hilbert cubes of squares. Asymptotic growth conjecture for three-dimension…

Let … be the set of integer squares, and let … be a Hilbert cube of dimension $3$. Existence conjecture for three-dimensional Hilbert cubes of squares. For each…

For an integer $m\geq 2$, let $S_m$ be the set of $m$-th powers, and let a Hilbert cube be a set of the form $H(a_0;a_1,\ldots,a_d)$. Folklore conjecture. For each $m\geq 2$, the d…

Chen–Lev strengthened conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exists an integer $l\ge1$ such that

Chen–Lev conjecture. If $R_C(n)=R_D(n)$ for every positive integer $n$, then there exist positive integers $d_{i_1}<d_{i_2}<\cdots$ belonging to $D$ such that

Let $V$ be the set of pure powers. Pillai's conjecture. For every fixed integer $h$, the equation $u-v=h$ has only finitely many solutions in distinct pure powers $u,v\in V$. The c…