Open-problem statements, with their sources attached.

BMO#2 formulation variant

Alternative statement of beaver_math_olympiad_problem_2_antihydra using set size comparison instead of a recurrent sequence b.

Does $p_n/m_n \to \infty$ for almost all $n$?

BMO#5

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (0, 5)$ and

Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?

BMO#8

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (10, 12)$ and

More generally, let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$. This problem asks whether $q(n, \log n) \ge (2 + \epsilon) \log n$ infinitely often.

Taking $n$ to be the product of primes between $\log n$ and $(2 + o(1)) \log n$ gives an example where

Can one prove that $q(n, \log n) < (1 - \epsilon) (\log n)^2$ for all large $n$ and some $\epsilon > 0$?

Let $\operatorname{lcm}(1, \dots, n)$ denote the least common multiple of $\{1, \dots, n\}$. Let $p_k$ be the $k$-th prime. Is it true that for all $k \geq 1$, $\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)$?

Is there a function $f$ with $f(n)\to\infty$ as $n\to\infty$ such that, for all large $n$, there is a composite number $m$ such that

Here $p(m)$ is the least prime factor of $m$.

Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$. Does:

converge?

Are there any odd weird numbers?

Are there infinitely many primitive weird numbers?

Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?

For each $n \in \mathbb{N}$ choose some $X_n \subseteq \mathbb{Z}/n\mathbb{Z}$. Let $B = \{m \in \mathbb{N} : \forall n, m \not\equiv x \pmod{n} \text{ for all } x \in X_n\}$. Must $B$ have a logarithmic density?

Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true that $\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2$ exists (and is finite)?

For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős.

See also [208].

Let $\alpha,\beta \in \mathbb{R}$. Is it true that$\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$? This is also known as the Littlewood conjecture.

Let $f$ be the asymptotic distribution function of $\varphi(n)/n$, so that for each $c \in [0,1]$, $f(c)$ is the natural density of $\{n : \varphi(n) < cn\}$. Is it true that there is no $x$ such that the derivative $f'(x)$ exists and is positive?

Is there an infinite set $A \subset \mathbb{N}$ such that for every $a \in A$, there is an integer n such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer, then $\frac{n_a}{a} → \infty$ as $a → ∞$?

Chowla's cosine problem*

If $A\subset \mathbb{N}$ is a finite set of positive integers of size $N > 0$ then is there some absolute constant $c>0$ and $\theta$ such that $\sum_{n\in A}\cos(n\theta) < -cN^{1/2}?$

Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$