19 problems

Dekking's conjecture. For every integer $n\geq 1$,

Bugeaud's conjecture. The Hurwitz constant of any real quadratic irrational is at most

For an irrational number $\xi$, define the deterministic random walk by … Call $r$ a record if $S_r(\xi)$ is different from every preceding partial sum, including $S_0(\xi)=0$. Van…

Let $[a_0,a_1,\ldots]$ be a continued fraction with $a_n\in\mathbb{Q}$ for all $n$. Suppose it converges simultaneously in $\mathbb{R}$ and $\mathbb{Q}_p$ to a real quadratic irrat…

Let $\alpha^{(p)}$ be a quadratic irrational in $\mathbb{Q}_p$ that has an image in $\mathbb{R}$. Consider its continued fraction obtained by Browkin I, Browkin II or Algorithm MR.…

Let $\alpha^{(p)}$ be a p-adic quadratic irrational, and consider its Ruban continued fraction. Capuano–Veneziano–Zannier conjecture. The Ruban continued fraction of a quadratic ir…

Let $[a_0,a_1,\ldots]$ be the Browkin I, Browkin II or Algorithm MR expansion of a $p$-adic quadratic irrational $\alpha^{(p)}$ that can be embedded in the real numbers. Let…

Let $p$ be a prime, and consider the continued fraction expansions produced by the new algorithms for quadratic irrationals in $\mathbb{Q}_p$. Periodicity conjecture. The continued…

Let $h\in\mathbb{Z}$ be even. Consider the $p$-adic continued fraction expansion produced by the Browkin II algorithm. Even-period Browkin II conjecture. For every even…

For every $Q obreaktextinmathbb N$ and $xiinmathbb R$, Zaharescu's theorem gives integers $p,q$ with $1leq qleq Q$ such that … In particular, this applies to $\xi=\phi$. Quadratic…

Let $p$ be a prime and let $t\geq1$, excluding the case $p=3$, $t=1$. A Browkin continued fraction (BCF) is a $p$-adic continued fraction expansion, and $\mathcal{Y}$ is the set of…

Grepstad–Kaltenböck–Neumüller conjecture. The transition point for the behavior of $\liminf_{N\to\infty}P_N(\beta)$ is $b=6$; equivalently, the liminf should change behavior at…

Period-length distribution conjecture. For each $i\in\mathbb{N}_{+}$, the limits

Transcendence conjecture. If $x$ and $y$ are positive integers with $x\neq y$, then $F(x,y)$ is transcendental.

Let $\omega$ be a quadratic irrational whose continued fraction has period $\ell\ge1$, and let $p_k/q_k$ be its rational convergents. Let…

A real number has a continued-fraction expansion … with $a_0\in\mathbb Z$ and $a_i\in\mathbb N$ for $i\geq 1$. For integers $p,q$ with $p^2+4q>0$, let…

Let $x_0=(u_0+v_0\sqrt{w_0})/z_0$ be a real root of $x^2+px+q$ with $p,q\in\mathbb Z$, and write the successive quantities produced by the continued-fraction algorithm as…

Average period conjecture. There is a constant such that

Quadratic-irrational discrepancy relation conjecture. Such a relation should hold for every pair $(\alpha,\beta)$ with $\alpha$ quadratic irrational and…