39 problems

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…

For an integer $n>2$, define the left factorial by … Kurepa's conjecture. One has … equivalently, for every odd prime $p>3$, … This is a classical open problem concerning the arith…

Brocard–Ramanujan conjecture. This equation has only three integer solutions. In 1876 Brocard, and independently in 1913 Ramanujan, asked for all integer solutions; the problem rem…

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…

Let $p$ be an odd prime. Consider the residue classes modulo $p$ represented by the factorial sequence $n!$ as $n$ varies. Factorial residue-class omission conjecture. About $p/e$…

For a positive integer $m$, the double factorial $m!!$ is defined by … and … A solution of $a_1!!\cdots a_t!!=n!!$ is called nontrivial when it is not one of the solutions arising…

Erdős's conjecture. The equation has only finitely many integer solutions.

Converse parity conjecture. If $\delta_{n-q,n}$ is odd, then

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 $k=t!-t$…

Let $a,b,c$ be integers satisfying $a!b!=c!$ with $a<b$, and say that the solution is class $k$ when $c-b=k$. Class 2 nonexistence conjecture. There are no class $2$ solutions; equ…

Let $a,b,c$ be integers satisfying $a!b!=c!$ with $a<b$, and call a solution nontrivial when $c-b>1$. Erdős's conjecture. The only nontrivial solution is … The problem is open even…

Let $b\ge 2$ be an integer, let $n$ be a positive integer, and write $s_b(m)$ for the sum of the base-$b$ digits of $m$. Let $\Lambda_n=\operatorname{lcm}(1,2,\ldots,n)$. Here…

Uniform boundedness conjecture. There exists a positive constant $C_r$, depending only on $r$, such that every degree-$r$ polynomial $P$ gives at most $C_r$ solutions. The paper no…

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 generalized conjecture. For every integer $k\geq 0$, the proportion of elements $y\in\mathbb{F}_p$ for which exactly $k$ positive integers $n$ satisfy

Stauduhar's conjecture.

Complete-solution conjecture. The only solutions $(r,s,n)$ with $n\geq4$ are

Complete-solution conjecture. The solutions of this equation are exactly the listed parameter tuples: $(n;r,s)=(12;\pm1,1)$ with $U_n=\pm144$; $(n;r,s)$ in

Finiteness conjecture. There are only finitely many solutions of this equation with $n\in\{6,12\}$, whether $\alpha,\beta$ are real or complex conjugates.

The factorial-power limit conjecture.

The composite factorial-plus-one bump conjecture. There are infinitely many such $n$ for which $A(n)<A(n+1)$; heuristically, this should occur when $n!+1$ is a semiprime with both…

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

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