11 problems

Miller et al.'s conjecture. There exists an infinite sequence of square-free integers

A real number $x\geq 0$ is digit-special if, for every integer base $b\geq 3$, its base-$b$ expansion omits at least one digit from $0,1,\dots,b-1$. Finiteness conjecture. There ar…

For an integer, say that its base-$b$ expansion contains solely the digits $0$ and $1$ when no other digit occurs. Folklore base-expansion conjecture. The only integers whose base-…

Logarithmic exceptional-set conjecture. Under these conditions,

For $\beta>1$, let $A_k(\beta)$ denote the set of positions of digit $1$ in the binary expansion of $\beta^k$. Long-arithmetic-progression conjecture. If $\log\beta/\log2$ is irrat…

Let $p,q$ be two distinct prime numbers. For each $k$ and each admissible integer $m$, let $d_{k,m}$ denote the number of digit ones among the first $m$ digits of the $q$-ary expan…

For each integer $k\geq 0$, write the binary expansion of $3^k$ as a sequence of digits in $\{0,1\}$, and let the positions of its digit $1$ be regarded as a subset of the nonnegat…

Let $m$ be the number whose base-$g$ expansion consists of $g$ digits equal to $1$, and let $g=p+1$ where $p$ is a prime number. Primitive-number conjecture. Then $m$ is the first…

Sun's conjecture. (I) There are at least $k$ non-zero digits in the base-$m$ expansion of

Let $x$ be a real irrational algebraic number and let $g\geqslant2$ be a positive integer. A real number is normal in base $g$ if it is simply normal in base $g^m$ for every…

Let $x$ be a real irrational algebraic number, let $g\geqslant3$ be a positive integer, and let $a$ be an integer satisfying $0\leqslant a\leqslant g-1$. The $g$-ary expansion of…