Open-problem statements, with their sources attached.

Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions.

Hickerson conjectured the largest solution the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, is 16!=14!5!2!.

Surányi was the first to conjecture that the only non-trivial solution to a!b!=n! is 6!7!=10!.

Is Erdos375Prop true?

Are there infinitely many $n$ such that ${2n\choose n}$ is coprime to $105$?

Is there some absolute constant $C > 0$ such that

for all $n$?

Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of

is $p$?

Let F(n) := \max\{m + p(m) \mid \textrm{m < n composite}\}\} where $p(m)$ is the least prime divisor of $m$. Is it true that $F(n)>n$ for all sufficiently large $n$?

Let F(n) := \max\{m + p(m) \mid \textrm{m < n composite}\}\} where $p(m)$ is the least prime divisor of $m$. Does $F(n) - n \to \infty$ as $n\to\infty$?

A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$

There is a $k$, such that $2 \le k \le n - 2$ and $\binom{n}{k}$ can be the product of consecutive primes infinitely often?

For all $2 \le k \le n - 2$, can $\binom{n}{k}$ be the product of consecutive primes infinitely often?

Can $\binom{n}{2}$ be the product of consecutive primes infinitely often?

The following is Schinzel's conjecture, which appears in [Gu04].

Is it true that for every $n \geq 1$ there is a $k$ such that

Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$?

Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?

Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?

Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$

Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$.