8 problems

Small-digit conjecture. There are infinitely many integers $n$ such that, for every $1\leqslant j\leqslant r$, all base-$g_j$ digits of $n$ are $<\kappa_jg_j$.

Conjecture on π and e. The numbers $c0$ and $e$ are multiplicatively independent; equivalently, $c0$ has no nonzero rational power equal to an integral power of $e$, or equivalentl…

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…

Let $p_1,\dots,p_k$ be strongly multiplicatively independent integers. For each $i$, let $D_i\subset\{0,\dots,p_i-1\}$ and define … Set $A=\bigcap_{i=1}^k A_{p_i}(D_i)$ and…

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…

Let $k>2$ be an integer, let $\zeta_k$ be a primitive $k$-th root of unity, and let $m>n>1$ be integers. Two algebraic integers are multiplicatively independent if the equation…

Multiplicative independence conjecture. The algebraic integers