7 problems

For a prime $p$, let $s_p(n)$ be the sum of the base-$p$ digits of $n$, let $\omega(n)$ count the distinct prime divisors of $n$, and define … Kellner's conjecture. The following s…

Fix a base $b\geq 2$. Write every $u\geq 1$ uniquely as a tower of powers of $b$ ending in $0<\omega\leq 1$, and let $\operatorname{ht}(u)$ be the height of this tower. Let $K(n)$…

Infinitude conjecture for the sets of digit-sum classes. For each $d\geq2$, the sets $\mathcal{S}_d$ and $\mathcal{S}\setminus\mathcal{S}_d$ are infinite. Moreover,…

Infinitude conjecture for primary Carmichael numbers. The following claims are true: the set $4\mathcal{C}'4$ is infinite, and the set $4\mathcal{C}\setminus\mathcal{C}'4$ is infin…

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x_1,\ldots,x_r,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the b…

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the binary digits…

Let $b,r$ be positive integers with $\gcd(b-1,r)=1$, and let $c$ be any integer. Write $s_b(n)$ for the sum of the base-$b$ digits of $n$. Bounded-offset conjecture. There exists a…