15 problems

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

Let … be the real number whose decimal expansion is obtained by listing all prime numbers in increasing order. A number is normal in base $s$ if every digit…

Small-digit conjecture. There are infinitely many integers $n$ such that, for every $1\leqslant j\leqslant r$, all base-$g_j$ digits of $n$ are $<\kappa_jg_j$.

Consider positive integers $n$ for which the central binomial coefficient $\binom{2n}{n}$ is coprime to $1155=3\times5\times7\times11$. 1155 conjecture. The only such integers are…

Let $p_1,\dots,p_k$ be strongly multiplicatively independent integers. For each $i$, let $D_i\subset\{0,\dots,p_i-1\}$ and define … Set $A=\bigcap_{i=1}^k A_{p_i}(D_i)$ and…

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-…

Let $n$ be an integer, and write $2^n$ in base $3$. Erdős's ternary problem. There are only finitely many integers $n$ such that the ternary expansion of $2^n$ does not contain the…

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, 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…