6 problems

Uniformity conjecture for singular $S$-units. For every $s \in \mathbb{N}$, the set $\mathcal{J}_s$ is finite.

Finiteness conjecture for singular S-units. For every $s \in \mathbb{N}$, the number of singular moduli that are $S$-units for some set of rational primes $S$ with $\#S=s$ is finit…

Let $k$ be a number field, let $S_{\infty}$ be the set of its archimedean places with $|S_{\infty}|\geq 2$, and let $\Lambda_{S_{\infty}}(k)$ be the logarithmic embedding, of rank…

Let $q$ be a prime. The $\{5,q\}$-Diophantine quadruple conjecture. If $q\not\equiv1\pmod4$, then no $\{5,q\}$-Diophantine quadruple exists. The authors say that this should be pro…

Let $p<q$ be primes and let $S=\{p,q\}$. An $S$-Diophantine quadruple is a quadruple of distinct positive integers such that every pairwise product plus one has all prime divisors…

Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all Archimedean places, and let $\operatorname{{\mathfrak o}}^*_S$ denote the group of $S$-units. For…