361 problems

Every rational number $a/q\in(0,1)$ has a continued-fraction expansion … where the $a_i\geq 1$ are its partial quotients. Zaremba's conjecture. There exists a constant…

Let $D\subseteq\mathbb{N}$ be an infinite set, and define … Let … be the exponent of convergence of the associated series. Hirst's conjecture. The Hausdorff dimension satisfies … H…

Hensley's conjecture. One has $\delta_A>1/2$ if and only if $D_A\supset\mathbb{N}_{\gg 1}$, meaning that Zaremba's conjecture holds for all sufficiently large $N$.

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…

Lubinsky's conjecture. The set $S$ is empty. This conjecture was disproved: Verschueren, and independently Grepstad, Kaltenböck and Neumüller, proved that the Golden Ratio…

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…

Texan conjecture. The dimension spectrum is full:

Stieltjes moment conjecture. The sequences of polynomials

Hensley's conjecture. For all sufficiently large $u$, there is a coprime integer $t<u$ such that

Let $f_n(x)$ be the polynomial sequence defined earlier in the paper, and let $a$ be a positive integer. Prime-value conjecture. For every positive integer $a$, there are infinitel…

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

Luzzi–Marmi conjecture. The entropy function $\alpha\mapsto h(T_\alpha)$ is continuous on the full parameter interval $(0,1)$.

Let $P$ be a $k$-term partition, and let $\accentset{\text{\large\bfseries.}}{K}(\cdot)$ denote the semi-regular continuant associated with an arrangement of the parts of $P$, whos…

Mordell's conjecture. The prime $p$ does not divide $y$.

Let $\lambda_n$ be the eigenvalues of the transfer operator for the Gauss continued fraction map, arranged so that $|\lambda_1|\geq |\lambda_2|\geq\cdots$, and let…

Niederreiter's conjecture. For the alphabet $\mathcal A=\{1,2,3\}$, every sufficiently large natural number occurs as a denominator:

Let $\{T_\alpha\}_{\alpha\in(0,1]}$ be the family of $\alpha$-continued fraction transformations. A parameter interval $I\subset(0,1]$ is a matching interval if there exist…

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…

Chow–Long's conjecture. The set $S_\alpha$ is uniquely avoidable.

Let $\mathscr{C}\subset\mathbb{R}^n$ be an irrational $n$-dimensional simplicial cone, let $K$ be the convex hull of the nonzero points of $\mathbb{Z}^n$ contained in $\mathscr{C}$…

Constancy conjecture. The entropy integral is constant and equal to $\frac{\pi^2}{6}$ for every $0<\alpha<\frac{1}{2}$.

Ramanujan's conjecture. If $a_i\to a>\frac14$, then the continued fraction diverges.