7 problems

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

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…

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.