Open-problem statements, with their sources attached.

Is it true that $A(x) \le x^{o(1)}$?

Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$. Does

exist?

Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$. What is

Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then does

exist?

Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then what is

Regarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and write "...if this trend conditions we expect [the limit] to be around 0.5, if it exists."

For all $n\ge 2k$ the least prime factor of $\binom{n}{k}$ is $\le\max(n/k,k)$, with only finitely many exceptions.

Ecklund, Erdős, and Selfridge [EES74] conjectured $g(k)\leq \exp((1+o(1))k)$.

Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.

Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$.

Is every odd $n > 1$ the sum of a squarefree number and a power of 2?

Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4.

Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?

1. There is NO good sequence with polynomial growth.

1. There is a good sequence with sub-exponential growth.

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.

Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.

Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?

For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?

Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?