6 problems

Erdős–Stewart–Tijdeman conjecture. For every $\epsilon>0$, there is $s_0(\epsilon)$ such that, for all $s>s_0(\epsilon)$, the number of $S$-unit solutions satisfies both

Let $S$ be a finite nonempty set of primes. Let $\mathcal{G}$ be the graph with vertex set $\mathbb{Q}$ in which two rationals are adjacent when their difference is an $S$-unit. A…

The abc-conjecture. For each positive real number $\varepsilon$, there is a positive number $C(\varepsilon)$ such that, whenever $a,b,c$ are coprime positive integers satisfying…

Let $S(n)$ denote the set of the first $n$ rational primes, and let $Sigma(S)$ be the set of solutions of the S-unit equation up to symmetry. The bounded-radical S-unit solution co…

Let $S$ be a finite set of primes with $\#S\geq2$, and let $\mathcal{G}$ be its $S$-unit graph. A cycle is nondegenerate if its edge differences have no proper nonempty zero subsum…

Let $S$ be a finite nonempty set of primes, let $\mathbb{Z}_S^*$ denote the group of $S$-units, and set … A positive $S$-unit is an element of $\mathbb{Z}_S^*$ that is positive. Ru…