5 problems

Let $\mathcal{R}$ be a commutative ring with unity and let $n\in\mathcal{R}\setminus\{0\}$. A Diophantine quadruple with property $D(n)$ in $\mathcal{R}$ is a set of four nonzero e…

Let $\mathbb{Z}[\omega]$ be the ring considered in the paper, and call a set of four distinct elements a $D(-1)$-quadruple if the product of any two distinct elements, decreased by…

Let $\{a,b,c,d\}$ be a Diophantine quadruple such that … Suppose that $s^2=ac+1$ and $t^2=bc+1$. Strengthened size conjecture. If … then … This strengthens the preceding lower boun…

Solitary powers conjecture. The element $p^k$ is $n$-powerfully solitary in $\mathcal O_{\mathbb{Q}(\sqrt{d})}$ for every positive integer $n$. More strongly,…

Let $k$ be the exponent appearing in Theorems 2.2 and 2.4 for the classification of $2$-powerfully perfect numbers in the relevant quadratic rings. Exponent-one conjecture. The val…