4 problems

Gurvich–Naumova conjecture. Every square pair $(k,j)$ is either one of the pairs $(k,j_i(k))$ for a positive integer $i$, or is a bi-Pythagorean pair of opposite parity.

Let $a\ge0$, $d\ge1$, and $N\ge1$ be integers, and define … Assume that $a\le\left(\frac{Nd-1}{2}\right)^2$ and $N>d+2\sqrt a$. The conjectural bounded-distance estimate. For every…

Let $a\ge0$, $d\ge1$, and $N\ge1$ be integers, and define … and … Assume that $a\le\left(\frac{Nd-1}{2}\right)^2$ and $N\le d+2\sqrt a$. The conjectural distance bound. For every…

Let $\textsc{Squares}$ denote the set of perfect squares in $\mathbb Z$. Finite-translate perfect-squares conjecture. There exists a positive integer $k$ such that there are no set…