20 problems

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,

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 $\mathscr{L}$ 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…

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…

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…

For each positive integer $n$, let $\mathscr{F}_n$ be the Fermat-type Calabi–Yau $n$-fold and let $t$ denote the mirror map for its Fermat pencil, with Fermat point $\psi=0$. Mirro…

Let $b\geq 2$ be the fixed integer base. Let $\epsilon>0$, let $k,l$ be positive integers, and let … … be polynomials with real algebraic coefficients, where $a_k$ and $c_l$ are li…

Let $\alpha$ be an algebraic number of degree $d\geq2$. Consider the rational vector space spanned by the logarithms of its integer shifts,…

For an algebraic number $\alpha$, consider the set ${\mathbb N}+\alpha=\{n+\alpha:n\in{\mathbb N}\}$. A set of nonzero algebraic numbers is multiplicatively independent if no nontr…

Algebraic-irrational conjecture. For every algebraic $\alpha\in\mathbb{R}\setminus\mathbb{Q}$,

Let $x$ be a real irrational algebraic number, let $g\geqslant3$ be a positive integer, and let $a$ be an integer satisfying $0\leqslant a\leqslant g-1$. The $g$-ary expansion of…

Let $r\in(0,1)$ be an irrational algebraic real, and let $p_r(m)$ denote the number of distinct binary words of length $m$ occurring in the binary expansion of $r$. Borel's conject…