16 problems

For a positive integer $A$, let … where the set is counted as a multiset when cardinalities are computed. Equidistribution conjecture. For any interval $I\subset[0,1]$, … for some…

Reducibility conjecture. If $x^{\underline{k}}-a!$ is reducible over $\mathbb{Z}$, then $(k,a)=(3,6)$, $(k,a)=(4,7)$, or there exists an integer $t$ such that $a=t!-1$ and $k=t!-t$…

For each integer $n\geq 2$, let the left factorial $\mathbf{!n}$ be the sum of the factorials below $n$, and let $n!$ denote the usual factorial. Kurepa's conjecture. The two facto…

Let $n_1,n_2,n_3,\ldots$ be the sequence defined by $\sigma_{n_\ell}=\ell+1$ for all $\ell\geq 1$, where the values of $\sigma_n$ arise from the paper's preceding construction. Quo…

Let $S^{(2)}_k$ denote the relevant sequence, let ${\rm Fail}(S^{(2)}_k)$ be its repair factor, and let ${\rm Rad}(n)$ denote the radical of $n$, the product of its distinct prime…

Stauduhar's conjecture.

The factorial-power limit conjecture.

The $D(n)$ bump conjecture. There are arbitrarily large $n$ such that

The $C(n)$ bump conjecture. There are arbitrarily large $n$ such that

The $B(n)$ bump conjecture. There are arbitrarily large $n$ such that

The $A(n)$ bump conjecture. There are arbitrarily large $n$ such that

For a positive integer $n$, let $S(n)$ be the smallest positive integer $j$ such that $n$ divides $j!$. Sondow's conjecture. The inequality … holds for almost all positive integers…

Prime factorization conjecture for factorials plus one. For every $n\in\mathbb N\setminus S$, the number $n!+1$ is square-free.

Let $p_n$ denote the $n$-th prime. Prime-index inequality conjecture. For all integers $n\geq 2$ and all integers $k$ satisfying $1\leq k\leq n-1$, … The inequality would imply tha…

Digit-discrepancy conjecture. The number of unequal digits between $Z_b(b^k)$ and the first $k$ digits of the base-$b$ expansion of $\vartheta(b)$ is smaller than or equal to the n…

I proved long ago that every $m < n!$ is the distinct sum of $n - 1$ or fewer divisors of $n!$. Let $h(m)$ be the smallest integer, if it exists, for which every integer less than…