Open-problem statements, with their sources attached.

Let V(x) count the number of n≤x such that ϕ(m)=n is solvable. Is there an asymptotic formula for V(x)?

Let$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$and$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$ Does $\lim V(x)/V'(x)$ exist?

Formalization note: We formalize the limit of the inverse fraction V'(x)/V(x) to ensure the limit is finite (bounded between 0 and 1).

Is it $>1$?

It is open whether the set of non-cototients has positive density.

Is there a sequence $1 \le d_1 < d_2 < \dots$ with density 1 such that all products $\prod_{u \le i \le v} d_i$ are distinct?

Does $f(n)$ miss infinitely many integers?

Is $f$ surjective?

How does $f$ grow?

Does $f$ become stationary at some point?

Let $a_1 = 2$ and $a_2 = 3$ and continue the sequence by appending to $a_1, \ldots, a_n$ all possible values of $a_i a_j - 1$ with $i \neq j$. Is it true that the set of integers which eventually appear has positive density?

How can the upper bound be improved?

Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0 < a < n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$

How can the lower bound be improved?

Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?

This is discussed in this MathOverflow question [MathOverflow].

What is the exact value of the constant?

Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?

BMO#1

Let $(a_n)_{n \ge 1}$ and $(b_n)_{n \ge 1}$ be two sequences such that $(a_1, b_1) = (1, 2)$ and

Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n. Must lim q n / (n ^ 2) = ∞?

BMO#2

Antihydra is a sequence starting at 8, and iterating the function $H(n) = \left\lfloor \frac {3n}2 \right\rfloor.$ The conjecture states that the cumulative number of odd values in this sequence is never more than twice the cumulative number of even values. It is a relatively new open problem with, so it might be solvable, although seems quite hard because of its Collatz-like flavor. The underlying Collatz-like map has been studied independently in the past, see doi:10.1017/S0017089508004655 (Corollary 4).

It is equivalent to non-termination of the 1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA 6-state Turing machine (from all-0 tape). Note that the conjecture that the machine does not halt is based on a probabilistic argument.

This machine and its mathematical reformulations were found by bbchallenge.org contributors mxdys and Rachel Hunter on June 28th 2024.

Is it true that $m_n<p_n$ for almost all $n$?