4 problems

For an integer $b\geq 3$, let $B_b$ be the set of positive integers whose base-$b$ expansion contains only the digits $0$ and $1$. Let $a,b$ be multiplicatively independent integer…

Let $b\geq 3$ be an integer, and let $B_b$ denote the set of positive integers whose base-$b$ expansion contains only the digits $0$ and $1$. Let $a,b,c\geq 3$ be pairwise multipli…

For an algebraic number $\alpha$, consider the set ${\mathbb N}+\alpha=\{n+\alpha:n\in{\mathbb N}\}$. A set of nonzero algebraic numbers is multiplicatively independent if no nontr…

Multiplicative independence conjecture. The algebraic integers