Open-problem statements, with their sources attached.

Is it true that $f(p)/p \to 0$ for $p \to \infty$ in a density 1 subset of the primes?

Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \le x$ for which $f(p)=p−1$ is $o(x/\log x)$.

[HaSu02] Hardy, G. E. and Subbarao, M. V., A modified problem of Pillai and some related questions. Amer. Math. Monthly (2002), 554--559.

Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?

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

Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$. Show that $f(n)=o(\log n)$.

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 $g(k)\leq \exp((1+o(1))k)$.

Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 5$?

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?

Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$

Is it true that $\lim_{N \to \infty} k(N) - (e - 1)N = \infty$?

1. There is NO good sequence with polynomial growth.