Open-problem statements, with their sources attached.

Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that

Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which

Is there and $A \subset \mathbb{N}$ is such that $\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$ exists and is $\ne 0$?

Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$ of length ≥ 4, with $(n, d) = 1$, be a perfect power?

Denote by $M(n, k)$ the least common multiple of the finite set $\{n+1, \dotsc, n+k\}$. Is it true that for all $m \geq n + k$, we get $M(m, k) \neq M(n, k)$?

Is $\sum_{n=2}^\infty \frac{1}{n!-1}$ irrational?

Is it true that, for all sufficiently large $n$, there exists some $k$ such that

where $p(m)$ denotes the least prime factor of $m$?

Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?

Erdős problem 681.* Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$, where $p(m)$ is the least prime factor of $m$ ?

There exists $c > 0$ such that $P(n, k) > \min\{n-k+1, k^{1 + c}\}$ for all $0 < k < n$.}

Standard heuristics suggest that $P(n, k) > e^{c\sqrt{k}}$ for some constant $c > 0$.

Can every integer $N≥2$ be written as $N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$?

Can every square $N≥2$ be written as $N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$?

Can $4$ be written as $4=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$?

Can $25$ be written as $25=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$ for some $k≥2$ and $m≥n+k$?

Estimate $\epsilon_n$ - lower bound.

Estimate $\epsilon_n$ - upper bound.

In particular, is it true that $\epsilon_n = o(1)$?

Let n be sufficiently large. Is there some choice of congruence class a_p for all primes 2 ≤ p ≤ n such that every integer in [1,n] satisfies at least two of the congruences ≡ a_p (mod p)?

Carmichael has asked whether there is an integer $n$ for which $\phi(m) = n$ has exactly one solution, that is \frac{f_\max(n)}{f_\min(n)} = 1.