41 problems

Let $j$ denote the involution of the real line induced by Dyer's outer automorphism of the extended modular group $\operatorname{PGL}(2,\mathbb{Z})$. The map $j$ preserves the set…

Let $(\alpha_1,\ldots,\alpha_n)$ be an $n$-tuple of real numbers in $\bigtriangleup$, and suppose that its triangle sequence is eventually periodic. Algebraicity conjecture for per…

Let $n$ be a positive integer and let $\bigtriangleup$ denote the region of ordered $n$-tuples used by the triangle algorithm. Let $0 \leq \alpha_n \leq \cdots \leq \alpha_1 < 1$ b…

Algebraicity conjecture. Every $c\in\mathcal{S}$ is either an integer or an irrational algebraic number; equivalently,

Let $\alpha$ be an algebraic integer of degree $d$, and consider the extremal algebraic integers for the house among the relevant nonreciprocal cases. Here nonreciprocal means that…

Let $A_d$ denote the set of real algebraic numbers of degree $d$. A real number is badly approximable if its rational approximations do not achieve arbitrarily small errors relativ…

Krachun–Petrov conjecture. For every finite subset $A\subset\mathbb{C}$,

Let $\alpha\in\mathbb{R}$ be an irrational algebraic number. A real number is normal if its digit frequencies are uniform in every integer base. Normality conjecture for irrational…

Let $A$ be a positive $m \times n$ matrix, let $V$ be a positive $m \times 1$ matrix, and let $W$ be a positive $1 \times n$ matrix whose entry sums agree. Let $x$ be the top-left…

Let $A$ be a general positive $3 \times 3$ matrix, and let $\operatorname{Sink}(A)$ denote its Sinkhorn limit. An entry is said to have degree over a field if it is algebraic over…

Let $A$ be a positive $m \times n$ matrix, and let $x$ be an entry of its Sinkhorn limit, obtained by iteratively scaling so that each row sum is $1$ and each column sum is…

Let $\mathscr{L}$ be the $\mathbf{Q}$-vector space of logarithms of algebraic numbers, and let $\mathscr{L}_p$ be its $p$-adic analogue. Consider an $n\times n$ matrix whose entrie…

An absorbing game has a limit value, and a rational absorbing game is one whose payoff and transition data are rational. A number is algebraic if it is a root of a nonzero polynomi…

Let $m\geq 1$. A rational absorbing game with $m$ actions per player is an absorbing game whose data are rational and in which each player has $m$ available actions. Maximal-degree…

Bounded-means conjecture. For every algebraic number $a$, the arithmetic means of its continued-fraction coefficients are bounded.

No-proper-large-solutions conjecture. No proper large solutions to the resulting equations exist, because each fixed algebraic number can have only finitely many large ABC hits.

Stronger approximation-speed conjecture. The product

Lang's conjecture. For every such sequence, the product

Let $\lambda$ be a real algebraic number. Write its minimal polynomial as $f(x)=\prod_{i=1}^d(a_i x+b_i)$ over the complex numbers, with coprime integer coefficients, and define…

Unboundedness conjecture. For every $N\in\mathbb N$, there exists an algebraic number having at least $N$ exceptional points.

Let $\alpha$ be an irrational algebraic number and let $n>1$ be an integer. An $n$-adic expansion of $\alpha$ is a representation $\sum_{l=0}^{\infty}a_ln^l$ with $0\le a_l<n$ sati…

Periodicity conjecture. The heuristic APD-algorithm is periodic for all triples of cubic conjugate vectors.

Equality conjecture. For any real algebraic $\alpha$, the inequality becomes an equality. For complex algebraic $\alpha$, the analogous equality holds for $A\subset\mathbb{C}$. Thi…

Algebraicity conjecture. All spherical codes in the Current Status of Unknown Parameters section are algebraic; in particular, their associated defining polynomials have finite deg…

Let $\alpha$ be a real algebraic number of degree at least $3$, and let $k>1$. Logarithmic rational-approximation conjecture. The inequality … has only finitely many rational solut…