Open-problem statements, with their sources attached.

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 $n$?

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.

Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 1.

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 $\epsilon > 0$?

For every $n$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$.

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 $k \geq 1$, $\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)$?

If $n \ge 2$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 1.

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 $m$.

If F is a decreasing family of sets of some finite type α, then there is some element x of α such that the family consisting of all members of F containing x is an intersecting subfamily of F with maximal cardinality.

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?

For even m > 2, it is open whether the cube digraph on (ZMod m)³ has a Hamiltonian arc decomposition.

Are there any odd weird numbers?

If x is a fusible number and y is its successor, then the interval [x + 1, y + 1) can be divided into intervals [ℓₙ, ℓₙ₊₁), such that the fusible numbers in [ℓₙ, ℓₙ₊₁) are obtained by fusing the n + 1st successor of x with a fusible number. This formalization differs from Conjecture 7.1 in the paper in four ways: (1) it is obtained from Conjecture 7.1 by plugging in n + 1 into n, which simplifies the expressions and removes the need to assume n ≥ 1; (2) the n + 1st successor s^(n+1)(x) is replaced by the explicit value x + (2 - 1 / 2 ^ n) * m; (3) instead of defining y to be the successor of x, we assert that there is no fusible number strictly between x and y; (4) instead of using ∃ z, IsFusible z ∧ q = s^(n+1)(x) ~ z we use the value of z determined by the equality, namely z = 2 * q - 1 - s^(n+1)(x), and it is easy to see z ∈ [x + 1 - m / 2 ^ n, x + 1) as required.

Are there infinitely many primitive weird numbers?