9 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…

Let $\mathcal{W}_2$ be the set of all positive integers $a$ for which there is no integer $m>1$ such that $(a-2^k,m)>1$ for every positive integer $k$. The $\mathcal{W}_2$ density…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. Let $n,r,s\geqslant 1$ with $r\not\equiv s\pmod 2$. Consecutive-index product congruence conjecture. … This…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r,s\geqslant 1$. Even-exponent Catalan-triangle congruence conjecture. … This is a complementary…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, let $r\geqslant 0$ and $s\geqslant 1$, and assume $n\geqslant 4^s-1$. Power-of-four threshold conjecture. ……

A positive power of two is an integer of the form $2^a$ with $a\in\mathbb{Z}^+$. Sun's prime–three-powers-of-two conjecture. (i) Every odd integer greater than $8$ can be written a…

Highest-prime-digit power-of-two conjecture. The number $32$ is conjectured to be the highest power of two whose decimal representation has all prime digits.