5 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…

Sharp-exponent conjecture. If $s>1$, then for each $\epsilon>0$ there exists a constant $C$ such that, for every integer $N$, the number of solutions $(x,y,z)$ with $x,y,z\geq 0$,…

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 $k\geq 2$ different odd prime numbers. For each $i\in\\{1,\dots,k\\}$, set … Define … Here $N_{p_1,\dots,p_k}^{B_1,\dots,B_k}$ denotes the set of nonnegative…

Let $q>2$, let $a_0\in\{0,\dots,q-1\}$, and let … be the set of nonnegative integers whose base-$q$ expansion contains no digit equal to $a_0$. Missing-digit prime conjecture. Ther…