10 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 $p\geq 5$, and let $\{|B_n|\}_{n\in\mathbb{N}}$ be the sequence of denominators of convergents for Browkin I, Browkin II and Algorithm MR p-adic continued fractions. Denominato…

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 $m\geq 1$, and let $1,\alpha_0^{(1)},\ldots,\alpha_0^{(m)}$ be $p$-adic numbers that are linearly dependent over $\mathbb{Q}$. The finite-or-periodic conjecture. The $p$-adic J…

Let $p$ be a prime, and let $D\in\mathbb{Z}$ be not a perfect square. A Browkin II continued fraction is a continued fraction generated by Browkin's second $p$-adic continued-fract…

Let $p$ be a prime. A nice Browkin continued fraction is the special class of Browkin continued fractions introduced by Capuano, Murru, and Terracini. The nice-continued-fraction c…

Let $p$ be a prime. A Browkin I continued fraction is a continued fraction generated by Browkin's first $p$-adic continued-fraction algorithm; a square root of an integer means…

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…