8 problems

For each positive integer $n$, write the ternary expansion of $2^n$ as a finite string. Individual-power digit-frequency conjecture. In the ternary expansion of each individual pow…

For each positive integer $n$, let $S_n$ be the base-$3$ expansion of $2^n$. For a fixed positive integer $k$, consider the aggregate frequencies of the length-$k$ strings occurrin…

For each positive integer $n$, write the base-$3$ expansion of $2^n$ as a finite string $S_n$, and let $F_d(N)$ be the aggregate frequency of the digit $d\in\{0,1,2\}$ among the st…

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…

Erdős's conjecture. For every $n\geq9$, the ternary expansion of $2^n$ contains the ternary digit $2$.

Second-kind Apéry congruence conjecture. The numbers $A^{(3)}_n$ satisfy the following congruences modulo $9$:

Ternary digit-count conjecture.

Long-period conjecture. For every $K>2/\log(3/2)$, one has $S_n^{(K)}=S_n$ for all sufficiently large $n$; in particular,