Open-problem statements, with their sources attached.

Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?

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 $A(x) \le x^{o(1)}$?

Is there an integer $m$ with $(m, 6) = 1$ such that none of $2^k \cdot 3^\ell \cdot m + 1$ are prime, for any $k, \ell \ge 0$?

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?

Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?

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?

Is it true that $f(N)\sim N^{1/3}$?

Originally asked to Erdős by Bose.

This is discussed in problem C11 of Guy's collection [Gu04].

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

More generally, Bose and Chowla [BoCh62] conjectured that the maximum size of $A\subseteq \{1,\ldots,N\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then $\lvert A\rvert \sim N^{1/r}.$

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."

Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?

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

Szabo asks whether the maximal $t$ is given by

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 $c>0$.