171 problems

Let … and write its Stieltjes continued fraction as … Let $s_{b,c}(n)$ denote the corresponding Stieltjes parameters. Generalized Rueppel parameter conjecture. Then … where $P(k)$…

For a perfect matching $M$ on $[2m]$, let ${\rm cr}(M)$ and ${\rm ne}(M)$ denote its numbers of crossings and nestings, respectively, and let ${\mathcal{M}}_{2m}$ be the set of per…

A continued fraction has partial numerators $a_n$ and partial denominators $b_n$, with the displayed initial terms determining the pattern. Continued fraction conjecture for…

For positive integers $a_1,\ldots,a_\ell\geq1$, define the continued fraction … For a positive integer $A$, let $Q_A$ be the set of denominators of reduced fractions in $(0,1)$ adm…

A Markov number is a positive integer occurring in a Markov triple, that is, a triple of positive integers satisfying the Markov equation. A Markov triple is ordered here as…

Let $\mathcal{A}\subseteq\mathbb{N}$ be finite, let $v$ be the primitive vector from the associated orbit, and let $L$ be a linear functional. Define … and let…

Cassels–Swinnerton-Dyer's conjecture. The following are equivalent:

For each non-empty finite set $\mathcal{I}\subset\mathbb{N}$, let … where $a_n(x)$ are the continued-fraction digits of the irrational number $x$. Texan conjecture. The set … is de…

Entropy-volume conjecture. For all $n\geq 3$ and all $\alpha\in(0,1)$,

Hensley's conjecture. The denominators contain every sufficiently large natural number if and only if the Hausdorff dimension exceeds one half:

Effective Lang conjecture. There exists an absolute constant $C$ such that, for every real cubic irrational number $x$, all partial quotients satisfy

Let $\xi$ be a real number defined by a series of the form … for arbitrary $p/q\in\mathbb{Q}$ and positive integer parameters $z_2,z_3,\ldots$. Transcendence conjecture. Every such…

Let $\bar{\bar \alpha}=\left(\alpha_{0},\alpha_{1},\alpha_{2}\right)$ be any $\mathbb{Q}$-basis of a real cubic number field. Let $\mathscr S\left(\bar{\bar \alpha}\right)$ denote…

Continued-fraction conjecture. The positive powers of $q$ follow the sequence $0,3,10,21,\dots$, given by $(k-1)(2k-1)$, while the negative powers follow $0,5,14,27,\dots$, given b…

Continued-fraction conjecture. The positive powers of $q$ in this continued fraction follow the sequence $0,1,6,15,\dots$, given by $k(2k-1)$, while the negative powers follow…

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…

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality consists of faces with integer affine types. Triangular-face conjecture. For…

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality has faces with integer distances to the origin. Unit-face-distance conjectur…

A torus decomposition arising from a periodic two-dimensional continued fraction of cubic irrationality has faces, each equipped with an integer distance to the origin. Face-distan…

A two-dimensional (or, more generally, $n$-dimensional) continued fraction is represented by a sail, whose facets have homeomorphic types, adjacency relations, integer volumes and…

Let $U$ be the constructed polygonal surface and let conditions 1–7 be the preceding geometric and arithmetic conditions imposed in the construction of a sail for a periodic contin…

Fix an integer $B\geq2$, and let $\mathcal{C}_B$ be the set of irrational numbers whose partial quotients are bounded in average by $B$. For each $n$, consider rational points…

Let $T_m(x)$ denote the Chebyshev polynomial of index $m$, and let $\ell$ be a non-zero multiple of $4$. Consider … A continued fraction is semi-specializable if it has the partial…

Let $R_N$ be the quantity defined in the paper, let $g(N)=\sqrt{R_N/2}$, and let $N_k$ and $Q_{2k+1}$ be the sequences defined above. Write $\operatorname{PQ}_j(g)$ for the $j$-th…