2 problems

For a positive integer $m$, write its ternary expansion $(m)_3$ using digits from $\{0,1,2\}$. Erdős's conjecture. There are only finitely many powers of $2$ whose ternary expansio…

Let $p$ and $q$ be distinct primes, let $b\in\{0,1,\dots,q-1\}$, and write $(p^n)_q$ for the base-$q$ expansion of $p^n$. Denote by $d_n(b)$ the number of occurrences of the digit…