7 problems

Let $b\ge 2$ be an integer, let $n$ be a positive integer, and write $s_b(m)$ for the sum of the base-$b$ digits of $m$. Let $\Lambda_n=\operatorname{lcm}(1,2,\ldots,n)$. Here…

Let $\mathsf s(n)$ be the number of ones in the binary expansion of a nonnegative integer $n$. For $t\geq 0$ and $j\in\mathbb Z$, define … where $\operatorname{dens} A$ denotes the…

Niven progression conjecture. There exist infinitely many pairs $(b,d)$ for which there is a $b$-Niven $d$-AP of length

Anti-Niven progression conjecture. There exist infinitely many $b$-anti-Niven $d$-APs of length $p-1$.

For $x\geq 1$, let the base-10 SanD numbers below $x$ be the ordered pairs $(a,b)$ satisfying $1\leq a<b\leq x$ and $b-a=s_{10}(ab)$, where $s_{10}$ is the decimal digital-sum func…

A base-10 SanD number is an ordered pair $(a,b)$ with $1\leq a<b$ satisfying $b-a=s_{10}(ab)$, and a SanD prime is a SanD number $(p,q)$ in which both entries are prime. Here…

Let $q_1,q_2,m_1,m_2$ be integers $\geq 2$ such that … For an integer $q\geq 2$, let $S_q(n)$ denote the sum of the digits of $n$ in base $q$. Gelfond's conjecture. There exists…