5 problems

Let $q>1$ be an integer, let $F$ be a finite set of primes that does not contain all the primes dividing $q$, and let $a$ be a positive integer. Let $(N_i)_{i\geq 0}$ satisfy…

Let $P_n(d)$ be the probability that the digit $d\in\{0,1,2,\dots,9\}$ occurs in primes of length less than $10^n$. Mahadee-Kamrujjaman conjecture. For every choice of digit $d$, t…

For an integer $r$, let the number of digits equal to $1$ in its base-$3$ expansion be counted with multiplicity. Binary-power ternary digit-count conjecture. There exists an $N$ s…

Let $q>1$ be an integer, let $F=\{p_1,\dots,p_k\}$ be a finite set of primes that does not contain all the primes dividing $q$, and let $a$ be a positive integer. Say that an integ…

Let $p$ be a prime, let $q$ be a base that is not a power of $p$, and write the base-$q$ expansion of $p^n$ as having $k_n=\lceil n\log_q p\rceil$ digits. Let $d_{j,n}$ denote its…