36 problems

For every real constant $C>0$, does there exist an integer $K\ge3$ such that there are infinitely many triples $(a,b,n)\in\mathbb N^3$ satisfying $a>0$, $b>0$, $n>0$, … and the den…

Let $f(n)$ be the minimal $m$ such that $n! = a_1\cdots a_k$ with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that $f(n)-2n \sim c\frac{n}{\log n}?$

Does there exist a function $f:\mathbb{N}\to\mathbb{R}$ with $f(r)\to\infty$ as $r\to\infty$ such that, for every $r\in\mathbb{N}$ with $r\geq1$, there are infinitely many…

Is it true that for every sufficiently small positive real $\varepsilon$ and all real constants $C,C'$ with $0<C<C'$, there exist natural numbers $a,b,n$ such that $n>0$,…

Let $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\textrm{ finite}\right\}.$ If $k\geq 2$, then does $A$ contain only finitely many $k$th powers? Does it contain only finitel…

If $n! = \prod_i p_i^{k_i}$ is the factorisation into distinct primes then let $h(n)$ count the number of distinct exponents $k_i$. Prove that there exists some $c>0$ such that…

Let $k\geq 2$. Does $(n+k)!^2 \mid (2n)!$ for infinitely many $n$?

For every finite set $S$ of natural numbers consisting entirely of primes, there are infinitely many $n\in\mathbb N$ such that, for every $p\in S$, the $p$-adic valuation of $n!$ i…

Let $p$ be a prime and $A_p = \{ k! \pmod{p} : 1\leq k<p\}.$ Is it true that $\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$

Let $\tau(n)$ denote the number of positive divisors of $n$, and define … What is the set of limit points of $(R_n)_{n\in\mathbb{N}}$ as $n\to\infty$? The claim is that this set is…

Are there only finitely many triples $(p,a,k)$, with $p$ prime, satisfying $(p-1)!+a^{p-1}=p^k$?

For which integers $a\geq 1$ and primes $p$ is there a finite upper bound on those $k$ such that there are $a=a_1<\cdots<a_n$ with $p^k \mid (a_1!+\cdots+a_n!)?$ If $f(a,p)$ is t…

Does the equation … have only finitely many solutions $(m,S)$, where $m\in\mathbb{N}$ and $S\subseteq\mathbb{N}$ is finite with every $a\in S$ positive?

For any $k\geq 2$ let $g_k(n)$ denote the maximum value of $(a_1+\cdots+a_k)-n$ where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$. Can one show that…

Is it true that there are no $n,x,y,k\in\mathbb{N}$ satisfying $xy>1$, $k>2$, and either … or …

Does the equation $n!+1=m^2$ have any solutions with $n,m\in\mathbb{N}$ other than $n=4,5,7$? Equivalently, is …

Let $f(n)$ denote the minimal $m\geq 1$ such that $n! = a_1\cdots a_t$ with $a_1<\cdots <a_t=a_1+m$. What is the behaviour of $f(n)$?

For each positive integer $n$, let $A(n)$ be the least value of $t+1$ for which there exist natural numbers $a_0,\ldots,a_t$ satisfying $a_0\le\cdots\le a_t\le n^2$ and … Prove tha…

Let $t(n)$ be the largest possible $a_1$ among factorizations $n!=a_1\cdots a_n$ with $a_1\leq\cdots\leq a_n$. Determine the second-order behavior of $t(n)$: in particular, beyond…

Show that the equation $n! = a_1!a_2!\cdots a_k!,$ with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$, has only finitely many solutions.

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 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…

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 $n_1,n_2,n_3,\ldots$ 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…